\label{thm}
Let $0\leq f\in C^1([0,1))$ satisfying $F(1)=+\infty$.
Assume that $0\le u\le 1$ is a  stable radial solution to \eqref{d-p}.
If $2\le n\le 6$, then $u$ is regular and satisfies
$$\frac 12 u_r(1)^2\le F(\|u\|_{L^\infty(B_1)})\le  Cu_r(1)^2.$$
Additionally, if $f$ is nondecreasing, then
 $$F(\|u\|_{L^\infty(B_1)})\le C,$$
where  $C>0$ is a universal constant.

This paper focuses on the issues associated with MEMS problem:
\begin{align}\label{d-p}
\left\{
\begin{aligned}
-\Delta u & = f(u)&\quad{\rm in}&\quad B_1 \setminus\{ 0\} \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}
where $B_1$ is the unit ball centered at the origin in ${\mathbb R}^n$($n\ge 2$), $0\le u\le 1$ is a stable radial solution, and $f:[0,1)\to [0,+\infty)$ is
$C^1$-function which blow-up at
$u=1$. Throughout the paper, for each $0\le t\le 1$ we define $F(t):=\int^t_0 f(s)\,ds$ and require that $F(1)=\int^{1}_0f(s)\,ds=+\infty$.

Recall that a radial solution $u$ of \eqref{d-p} is called stable if
$$\int_{B_1}f'(u)\xi^2\,dx\le \int_{B_1}|\nabla \xi|^2\,dx$$
for every $\xi \in C^\infty(B_1)$ with compact support in $B_1\backslash\{0\}$, where
$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if
$\|u\|_{L^\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\'e \cite{bc}.
Notably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\ge 7$ and $f(t)=\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies
$\|u\|_{L^\infty(B_1)}=1$. In contrast, for dimensions $2\le n\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\'e \cite{bc} and Meadows \cite{m04}.

This example naturally leads to the following question: in dimensions $2\le n\le 6$, is every stable radial solution of \eqref{d-p} regular?

In the radial case, we provide a complete answer to this question.

\begin{align}\label{d-p}
\left\{
\begin{aligned}
-\bdz u & = f(u)&\quad{\rm in}&\quad B_1 \setminus\{ 0\}  \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}

Recall that a radial solution $u$ of \eqref{d-p} is called stable if
$$\int_{B_1}f'(u)\xi^2\,dx\le \int_{B_1}|\nabla \xi|^2\,dx$$
for every $\xi \in C^\fz(B_1)$ with compact support in $B_1\backslash\{0\}$, where
$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if
$\|u\|_{L^\fz(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\'e \cite{bc}.
Notably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\ge 7$ and $f(t)=\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies
$\|u\|_{L^\fz(B_1)}=1$. In contrast, for dimensions $2\le n\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\'e \cite{bc} and Meadows \cite{m04}.

This example naturally leads to the following question: in dimensions $2\le n\le 6$, is every stable radial solution of \eqref{d-p} regular?

(ii) When dimension
$3\le n\le 6$, $F(1)=+\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in Bruera and Cabr\'e \cite{bc}. Indeed, Bruera and Cabr\'e \cite{bc}
constructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\ge 3$ and for some
 $p\in (0,1)$ satisfying
$$-\bdz u=\frac{2}{1+p}\left(\frac{2}{1+p}+n-2\right)(1-u)^{-p}$$
with $F(1)<+\fz$. For dimension $n=2$ (without assuming $F(1)=+\infty$), Luo, Ye and Zhou \cite{lyz} showed that the stable radial solution $u$ of $-\bdz u+ c(x)\cdot \nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\Delta u = f(u)$.

Recently, inspired by a key stability inequality established \cite{cfrs},
Bruera and Cabr\'e \cite{bc} investigated the case of a nonlinearity $f(u)$
that blows up as $u\to 1^{-}$. Specifically, under the following Crandall-Rabinowitz type condition:
\begin{align}\label{gz}
\gz:=\liminf_{t\to 1^{-}}\frac{f(t)f''(t)}{f'(t)^2}>1
\end{align}
for all $f\in C^2([0,1))$ satisfying $f\ge 0$,
$f'\ge 0$ and $F(1)=+\fz$, they established the following local estimate for all stable solutions
(not necessarily radial) up to optimal dimension:
$$\|u\|_{L^{\fz}(B_{1/2})}\le F^{-1}(C\|u\|^2_{L^1(B_1)})<1\quad
\forall \, 2\le n\le 6,$$
where constant $C$ depending only on $n$ and $\gz$. Additionally, a forthcoming study by Figalli and Franceschini \cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.

In order to show Theorem \ref{thm}, we recall the following important result concerning stable radial solutions, which was established by Villegas \cite[Theorem 1.7]{v12}.
\begin{lem}\label{vi-lem}
Let $n\geq 2$ and $f:[0,1)\to [0,+\fz)$ be a $C^1$ function satisfying $f\ge 0$. Let $0\le u\le 1$ be a stable radial solution to
\eqref{d-p} in $W^{1,2}(B_1)$. Then there exists a constant $K_n$ depending only on $n$ such that
\begin{align}\label{vi-le1}
|u_r(t)|\le K_n \|\nabla u\|_{L^2(B_1\backslash B_{1/2})}t^{-n/2+\sqrt{n-1}+1},
\quad \forall \, 0<t<1/2.
\end{align}
\end{lem}
Before applying Lemma \ref{vi-lem}, we need to verify that
$u$ belongs to $u\in W^{1,2}(B_1)$. For this purpose, we state the following proposition.
\begin{prop}\label{w12}
Let $n\ge 2$ and $f:[0,1)\to[0,+\infty)$ be a $C^{1}$ function satisfying $f\ge 0$. Suppose that $0\leq u\leq 1$ is a classical nontrivial $C^{2}$ radial solution (not necessarily stable) of $-\bdz u = f(u) \quad{\rm in} \quad B_1\backslash\{0\} $. Then

As a direct consequence of Lemma \ref{vi-lem} and Proposition \ref{w12}, we have:
\begin{prop}\label{key-prop-2}
Let $n\geq 2$ and $f:[0,1)\to [0,+\fz)$ be a $C^1$ function satisfying $f\geq 0$. Let $0\le u\le 1$ be a stable radial solution to
\eqref{d-p}. Then there exists a constant $C_n$ depending only on $n$ such that

\noindent which completes the proof.
\end{proof}
In addition, if $f$ is nondecreasing, one can derive the universal bound
of $u_r(1)$, which is useful for establishing an upper bound of $F(\|u\|_{L^\fz(B_1)})$.
\begin{lem}\label{un-bd}
Let $0\le f\in C^1([0,1])$ be nondecreasing. Assume
that $0\le u\le 1$ is a stable radial solution to
\eqref{d-p}. Then
$$|u_r(1)|\le 2.$$

Finally, by $u\in C^1(\overline B_1\backslash B_{1/2})$, one gets
$F(\|u\|_{L^\fz(B_1)})<+\fz$ and so that $u$ is regular. Furthermore,
if $f$ is nondecreasing, Lemma \ref{un-bd} tells us $|u_r(1)|\le 2$, and thus
$$F(\|u\|_{L^\fz(B_1)})\le M_n u_r(1)^2\le 4M_n.$$
Since we are considering a finite
number of dimensions ($2\le n\le 6$), we can take $4M_n\leq C$ in order to
obtain a universal constant $C>0$.
Hence we finish this proof.
\end{proof}
We apply now Theorem \ref{thm} to prove Theorem
\ref{thm-2}.
\begin{proof}[Proof of Theorem \ref{thm-2}]
Let $0<\lz<\lz^{\star}$ and let $u_{\lz}\in C^2(\overline B_1)$ be a stable radial
solution to \eqref{g-dp}. It follows from Theorem \ref{thm} that
$$F(\|u_{\lz}\|_{L^\fz(B_1)})\le C(n)
\left(\frac{d u_{\lz}(1)}{dr}\right)^2.$$
Using Lemma 2.3 in \cite{v12} by Villegas, one has that
$$F(\|u_{\lz}\|_{L^\fz(B_1)})\le C(n)
\left(\frac{d u_{\lz}(1)}{dr}\right)^2
\le C(n)\left(\frac{d u_{\lz}(1/2)}{dr}\right)^2
\le C(n)\int^{3/4}_{1/2}\left(\frac{d u_{\lz}(t)}{dr}\right)^2\,dt,$$
where we used the fact that $t^{2n-2}\left(\frac{d u_{\lz}(t)}{dr}\right)^2$
is nondecreasing in the last inequality. Since
$\lim_{\lz\to \lz^{\star}}u_{\lz}=u^{\star}$ and
$\nabla u_{\lz}\to \nabla u^{\star}$ in $L^2(B_{3/4})$ as
$\lz \to \lz^{\star}$, passing to the limit $\lz\to \lz^{\star}$ we conclude that
$$F(\|u^{\star}\|_{L^\fz(B_1)})
\le C(n)\int^{3/4}_{1/2}\left(\frac{d u^{\star}(t)}{dr}\right)^2\,dt<+\fz.$$

\begin{align}\label{d-p}
\left\{
\begin{aligned}
-\bdz u & = f(u)&\quad{\rm in}&\quad B_1 \setminus\{ 0\}  \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}

\begin{align}\label{g-dp}
\left\{
\begin{aligned}
-\bdz u & = \lz f(u)&\quad{\rm in}&\quad B_1  \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}

\begin{thm}\label{thm}
Let $0\leq f\in C^1([0,1))$ satisfying $F(1)=+\fz$.
Assume that $0\le u\le 1$ is a  stable radial solution to \eqref{d-p}.
If $2\le n\le 6$, then $u$ is regular and satisfies
$$\frac 12 u_r(1)^2\le F(\|u\|_{L^\fz(B_1)})\le  Cu_r(1)^2.$$
Additionally, if $f$ is nondecreasing, then
 $$F(\|u\|_{L^\fz(B_1)})\le C,$$
where  $C>0$ is a universal constant.
\end{thm}

\begin{thm}\label{thm-2}
Let $f\in C^1([0,1))$ be nondecreasing, convex, and
$F(1)=+\fz$. Suppose that $0\le u^{\star}\le 1$ is the extremal solution to
\eqref{g-dp}. Then
$$\|u^{\star}\|_{L^\fz(B_1)}<1\quad \forall \, 2\le n\le 6.$$

\end{thm}

\begin{lem}\label{un-bd}
Let $0\le f\in C^1([0,1])$ be nondecreasing. Assume
that $0\le u\le 1$ is a stable radial solution to
\eqref{d-p}. Then
$$|u_r(1)|\le 2.$$

\end{lem}

We now provide some remarks on the main results.
\begin{rem}
\rm
(i) The dimension $n \leq 6$ in Theorem \ref{thm} is optimal: for $n \geq 7$ and $f(t) = \frac{n-1}{1-t}$, the function $1 - |x|$ is always stable but not regular. Additional examples can be found in Bruera and Cabr\'e \cite{bc}.

(ii) When dimension
$3\le n\le 6$, $F(1)=+\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in  Bruera and Cabr\'e \cite{bc}. Indeed, Bruera and Cabr\'e \cite{bc}
constructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\ge 3$ and for some
 $p\in (0,1)$ satisfying
$$-\Delta u=\frac{2}{1+p}\left(\frac{2}{1+p}+n-2\right)(1-u)^{-p}$$
with $F(1)<+\infty$. For dimension $n=2$ (without assuming $F(1)=+\infty$), Luo, Ye and Zhou \cite{lyz} showed that the stable radial solution $u$ of $-\Delta u+ c(x)\cdot \nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\Delta u = f(u)$.

(iii) In Theorem \ref{thm} above, with slight modifications, it can be shown that the stable radial  solution is regular under the conditions $0\le u\le 1$, $0\le f$  and $F(1)=+\infty$,
without requiring the boundary condition: $u=0$ on $\partial B_1$.
\end{rem}

Let us now review some relevant work in this direction. In the non-radial setting with a nonlinearity $f(u)$ that is unbounded as
$u\to +\infty$, a pioneering study was first conducted by Crandall and Rabinowitz \cite{cr75}
 with
$f(t)=e^t$ and $f(t)=(1+t)^p$, for $p>1$. Subsequently, the boundedness of stable solutions
$u$ to $-\Delta u=f(u)$ in general domains $\Omega$ has been extensively investigated over the past three decades; see \cite{cc06,c10,n20,v13}. Notably, Cabr\'e, Figalli, Ros-Oton and Serra \cite{cfrs} made a breakthrough:
Suppose $f$ is locally Lipschitz, nonnegative, and nondecreasing, and $u\in W^{1,2}_0(\Omega)$
is stable solution of $-\Delta u=f(u)$ in $C^3$ domain $\Omega$, they showed that
$$\|u\|_{C^{0,\alpha}(\overline \Omega)}\le C(n)\|u\|_{L^\infty(\Omega)}\quad
2\le n\le 9$$
for some $\alpha=\alpha(n)\in (0,1)$. They also established the local H\"older continuity of
$u$ for $n\le 9$. Note that the dimension $n\le 9$ is optimal for nonlinearities
$f(u)$ that blow-up as $u\to \infty$. In addition, Cabr\'e provided a quantitative proof of these results in \cite{c23,c24}.
We also mention that, as established in
the work of Erneta \cite{e23},
the optimal boundary regularity of $u$ in $C^3$ domains attainable already in the class of
$C^{1,1}$-domains.

Recently, inspired by a key stability inequality established \cite{cfrs},
Bruera and Cabr\'e \cite{bc} investigated the case of a nonlinearity $f(u)$
that blows up as $u\to 1^{-}$. Specifically, under the  following Crandall-Rabinowitz type condition:
\begin{align}\label{gz}
{\gamma}:=\liminf_{t\to 1^{-}}\frac{f(t)f''(t)}{f'(t)^2}>1
\end{align}
for all  $f\in C^2([0,1))$ satisfying $f\ge 0$,
$f'\ge 0$ and $F(1)=+\infty$, they established the following local estimate for all stable solutions
(not necessarily radial) up to optimal dimension:
$$\|u\|_{L^{\infty}(B_{1/2})}\le  F^{-1}(C\|u\|^2_{L^1(B_1)})<1\quad
\forall \,  2\le n\le 6,$$
where constant $C$ depending only on $n$ and ${\gamma}$. Additionally, a forthcoming study by Figalli and Franceschini \cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.

On the other hand, for smooth domains $\Omega$, Bruera and Cabr\'e \cite{bc} also derived global estimates for stable solutions
$u\in C^2( \Omega)\cap C^0(\overline \Omega)$  of the Dirichlet boundary value problem:
\begin{align*}
\left\{
\begin{aligned}
-\Delta u & = f(u)&\quad{\rm in}&\quad \Omega  \\
u& =0 \quad&{\rm on}&\quad \partial \Omega,\\
\end{aligned}
\right.
\end{align*}
and showed the following:
\begin{itemize}
\item[$\bullet$] If $n\le 2$, then $\|u\|_{L^\infty(\Omega)}<1$ without condition \eqref{gz}.

\item[$\bullet$] If $3\le n\le 6$ and $f$ satisfies the condition \eqref{gz}, then $\|u\|_{L^\infty(\Omega)}<1$.

\end{itemize}
It is worth noting that additional partial results related to this direction can be found in
\cite{ces,cgg,lyz,m04}. In particular, for dimension $n=2$ (without requiring $F(1)=+\infty$), Luo, Ye and Zhou \cite{lyz} proved that the stable radial solution $u$ of $-\Delta u+c(x)\cdot \nabla u=f(u)$
 is regular provided that $f$ is nonnegative, nondecreasing, and convex.  Motivated by these findings, Bruera and Cabr\'e \cite{bc} proposed the following open problem:
\begin{center}
{\bf Open problem}. Does regularity of stable solutions hold up to the optimal
dimension for singular nonlinearities under no Crandall-Rabinowitz type condition?
\end{center}
In the radial setting,
we provide a positive answer to this open problem in
Theorem \ref{thm}. However, the non-radial case remains open.

Theorem \ref{thm} has an important application to the Gelfand-type problem; see for example
in \cite{bv97,b03}. Precisely, let $f\in C^1([0,1))$ satisfy $f(0)>0$ and be nondecreasing, convex, and $F(1)=+\infty$.

Given a constant $\lambda>0$ consider the nonlinear elliptic problem:
\begin{align}\label{g-dp}
\left\{
\begin{aligned}
-\Delta u & = \lambda f(u)&\quad{\rm in}&\quad B_1  \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}
A fundamental question is to determine a  constant $\lambda^{\star}>0$ such that the problem
\eqref{g-dp} has a unique $L^1$-solution $u^{\star}\in L^1(B_1)$ in the sense that
$$-\int_{B_1}u^{\star}\Delta \xi \,dx=
\lambda^{\star}\int_{B_1}f(u^{\star})\xi\,dx\quad\forall \,  \xi\in C^{0,1}_c(B_1)$$
Such $L^1$-solution solution $u^{\star}$ is called extremal solution.
The uniqueness and existence of extremal solution were established by
Castorina, Esposito and Sciunzi \cite{ces}. Moreover, they
showed that  for every $0<\lambda<\lambda^{\star}$, there exist a stable solution
$u_{\lambda}\in C^2(\overline B_1)$ such that
$$\lim_{\lambda\to \lambda^{\star}}u_{\lambda}
=u^{\star}.$$
Combining this result with Theorem \ref{thm}, we conclude that the extremal solution is always regular in dimensions $2\le n\le 6$.

\begin{thm}\label{thm-2}
Let $f\in C^1([0,1))$ be nondecreasing, convex, and
$F(1)=+\infty$. Suppose that $0\le u^{\star}\le 1$ is the extremal solution to
\eqref{g-dp}. Then
$$\|u^{\star}\|_{L^\infty(B_1)}<1\quad \forall \, 2\le n\le 6.$$

\end{thm}

Finally, we outline the main ideas for proving the theorems.
Since Theorem \ref{thm-2} follows directly from Theorem \ref{thm},
it suffices to explain the proof strategy for Theorem \ref{thm}.

A key starting point is the following observation for general radial solutions:
$$
F(\|u\|_{L^\infty(B_1)})= (n-1)\int^1_0\frac{u_r(t)^2}{t}\,dt+\frac{u_r(1)^2}2.
$$
Since $u\in C^1(\overline B_1\backslash\{0\})$,
the main task is reduced to bounding the term $\int^1_0\frac{u_r(t)^2}{t}\,dt$. By a key
estimate due to Villegas \cite[Theorem 1.7]{v12} (see also \cite{cc06}), we deduce
\begin{align*}
\vert u_r(t)\vert \le C_n \vert u_r(1)\vert t^{-n/2+\sqrt{n-1}+1},
\quad \forall \, 0<t\leq 1.
\end{align*}
Observe that
$$-n/2+\sqrt{n-1}+1>0\quad{\rm if \ and\ only\ if}\quad 2\le n\le 6.$$
This implies that $u_r(t)$ is H\"older continuous near  $t=0$ and hence
the behavior of
 $t^{-1}u_r(t)^2$ is like  $t^{-\alpha}$ for some
$\alpha\in (0,1)$ whenever $2\le n\le 6$. Therefore, the integral
$\int^1_0\frac{u_r(t)^2}{t}\,dt$ is finite, and we conclude

$$F(\|u\|_{L^\infty(B_1)})\leq C_nu_r(1)^2<+\infty.$$
Combining this with the condition $F(1)=\infty$, we obtain $\|u\|_{L^\infty(B_1)}<1$.
Moreover, if $f$ is nondecreasing, the universal bound on $F(\|u\|_{L^\infty(B_1)})$ follows
from the upper bound on $|u_r(1)|$; see Lemma \ref{un-bd} for details.

To prove Theorem~\ref{thm}, since Theorem~\ref{thm-2} follows directly from it, the strategy starts from the identity for general radial solutions
\[
F(\|u\|_{L^\infty(B_1)})= (n-1)\int^1_0\frac{u_r(t)^2}{t}\,dt+\frac{u_r(1)^2}2.
\]
Thus (with $u\in C^1(\overline B_1\backslash\{0\})$) the main task is to bound $\int^1_0\frac{u_r(t)^2}{t}\,dt$. Using Villegas' estimate \cite[Theorem 1.7]{v12} (see also \cite{cc06}),
\[
|u_r(t)|\le C_n|u_r(1)|\, t^{-n/2+\sqrt{n-1}+1},\qquad 0<t\le 1.
\]
Since $-n/2+\sqrt{n-1}+1>0$ iff $2\le n\le 6$, this gives H\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\alpha}$ for some $\alpha\in(0,1)$ when $2\le n\le 6$. Therefore $\int^1_0\frac{u_r(t)^2}{t}\,dt<\infty$, yielding
\[
F(\|u\|_{L^\infty(B_1)})\le C_n u_r(1)^2<+\infty.
\]
Combining this with $F(1)=\infty$ gives $\|u\|_{L^\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\|u\|_{L^\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$ (see Lemma~\ref{un-bd}).

To prove the main theorem, since
\begin{thm}\label{thm-2}
Let $f\in C^1([0,1))$ be nondecreasing, convex, and
$F(1)=+\fz$. Suppose that $0\le u^{\star}\le 1$ is the extremal solution to
\eqref{g-dp}. Then
$$\|u^{\star}\|_{L^\fz(B_1)}<1\quad \forall \, 2\le n\le 6.$$

\end{thm}
follows directly from it, the strategy starts from the identity for general radial solutions
\[
F(\|u\|_{L^\infty(B_1)})= (n-1)\int^1_0\frac{u_r(t)^2}{t}\,dt+\frac{u_r(1)^2}2.
\]
Thus (with $u\in C^1(\overline B_1\backslash\{0\})$) the main task is to bound $\int^1_0\frac{u_r(t)^2}{t}\,dt$. Using Villegas' estimate \cite[Theorem 1.7]{v12} (see also \cite{cc06}),
\[
|u_r(t)|\le C_n|u_r(1)|\, t^{-n/2+\sqrt{n-1}+1},\qquad 0<t\le 1.
\]
Since $-n/2+\sqrt{n-1}+1>0$ iff $2\le n\le 6$, this gives H\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\alpha}$ for some $\alpha\in(0,1)$ when $2\le n\le 6$. Therefore $\int^1_0\frac{u_r(t)^2}{t}\,dt<\infty$, yielding
\[
F(\|u\|_{L^\infty(B_1)})\le C_n u_r(1)^2<+\infty.
\]
Combining this with $F(1)=\infty$ gives $\|u\|_{L^\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\|u\|_{L^\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$. We use the following lemma.
\begin{lem}\label{un-bd}
Let $0\le f\in C^1([0,1])$ be nondecreasing. Assume
that $0\le u\le 1$ is a stable radial solution to
\eqref{d-p}. Then
$$|u_r(1)|\le 2.$$

\end{lem}

\label{thm}
Let $0\leq f\in C^1([0,1))$ satisfying $F(1)=+\infty$.
Assume that $0\le u\le 1$ is a  stable radial solution to
\begin{align}\label{d-p}
\left\{
\begin{aligned}
-\bdz u & = f(u)&\quad{\rm in}&\quad B_1 \setminus\{ 0\}  \\
u& =0 \quad&{\rm on}&\quad \partial B_1,\\
\end{aligned}
\right.
\end{align}
If $2\le n\le 6$, then $u$ is regular and satisfies
$$\frac 12 u_r(1)^2\le F(\|u\|_{L^\infty(B_1)})\le  Cu_r(1)^2.$$
Additionally, if $f$ is nondecreasing, then
 $$F(\|u\|_{L^\infty(B_1)})\le C,$$
where  $C>0$ is a universal constant.,

Question:
Let $B_1\subset \mathbb{R}^n$ be the unit ball, let $f:[0,1)\to[0,\infty)$ be a $C^1$ function, and define
$$F(t):=\int_0^t f(s)\,ds.$$
Assume $F(1)=+\infty$, $2\le n\le 6$, and that $u$ is a radial function with $0\le u\le 1$ solving
$$\begin{cases}
-\Delta u=f(u) & \text{in } B_1\setminus\{0\},\\
u=0 & \text{on } \partial B_1,
\end{cases}$$
where $u$ is stable in the sense that
$$\int_{B_1} f'(u)\,\xi^2\,dx\le \int_{B_1} |\nabla \xi|^2\,dx$$
for every $\xi\in C^\infty(B_1)$ compactly supported in $B_1\setminus\{0\}$. Here $u_r(1)$ denotes the radial derivative at $r=1$, and “$u$ is regular” means $\|u\|_{L^\infty(B_1)}<1$. Which of the following statements is valid?

The solution $u$ is regular, and it satisfies
$$\frac12\,u_r(1)^2\le F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C\,u_r(1)^2.$$
Moreover, if $f$ is nondecreasing, then
$$F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C,$$
for a universal constant $C>0$.

Choices:
B.
The solution $u$ is regular, and it satisfies
$$\frac12\,u_r(1)^2\le F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C\,u_r(1)^2.$$
Moreover, if $f$ is nondecreasing, then the same conclusion holds with a universal constant $C>0$ for every dimension $n\ge 2$.
C.
The solution $u$ is regular, and it satisfies
$$F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C\,u_r(1)^2.$$
D.
The solution $u$ is regular, and it satisfies
$$\frac12\,u_r(1)^2\le F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C\,u_r(1)^2.$$
Moreover, if $f$ is nondecreasing, then
$$F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C,$$
where $C>0$ is a constant depending only on $n$ and not on $f$ or $u$.
E.
If $f$ is nondecreasing, then every stable radial solution $u$ satisfies
$$\|u\|_{L^\infty(B_1)}<1$$
for all $n\ge 2$, and moreover
$$\frac12\,u_r(1)^2\le F\bigl(\|u\|_{L^\infty(B_1)}\bigr)\le C\,u_r(1)^2.$$

dimension threshold from integrability exponent

dropped lower bound and dropped universal nondecreasing conclusion

dependence of the universal constant in the monotone case

uses monotonicity to bypass the sharp restriction $2\le n\le 6$

The stem does not explicitly state the conclusion in choice A, nor does it contain obvious wording cues that uniquely identify the correct option. The hypotheses are substantial, but they do not directly leak the exact form of the final estimates.

The item is close to a theorem-recall question: it presents the full hypothesis set and asks for the valid conclusion. However, it is not a pure tautological restatement, since the choices differ in meaningful ways (dimension range, sharpness of inequalities, and dependence/universality of constants).

Some reasoning is needed to distinguish the strongest correct conclusion from nearby variants, especially regarding the role of the dimension bound, the lower bound involving u_r(1), and universality of constants. Still, the task mainly tests recognition of the precise theorem statement rather than deeper derivation.

The distractors are mathematically plausible and target common failure modes: overextending the dimension range, weakening the conclusion, incorrectly changing constant dependence, or swapping the structure of the estimates. They are distinct and well aligned with how a learner might misremember the result.

A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than genuine generative mathematical reasoning.


Let $1<p<\infty$ and let $\Omega\subseteq\mathbb{R}^N$ be an open set. Let us suppose that there exists $k\in\mathbb{N}\setminus\{0\}$ such that
\begin{equation}\
\label{PL}
\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega).
\end{equation}
Then, for every $\ell\in\{1,\dots,k\}$ the minmax value $\lambda_{\ell,p}^{\rm LS}(\Omega)$ is an eigenvalue of the Dirichlet $p-$Laplacian on $\Omega$.
\par
Moreover, we have:
\begin{itemize}
\item if $\mathcal{E}_p(\Omega)<+\infty$,  for every $\ell\in\{1,\dots,k\}$ there exist two constants $C_\ell=C_\ell(N, p, \Omega)>0$ and $\alpha_\ell=\alpha_\ell(p,\Omega)>0$ such that for every eigenfunction $u_\ell\in W^{1,p}_0(\Omega)$ associated to $\lambda_{\ell,p}^{\rm LS}(\Omega)$, we have
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha_\ell\,|x|}\,,\qquad \text{for every}\ x\in\Omega;
\]
\item if $\mathcal{E}_p(\Omega)=+\infty$, for every $\ell\in\mathbb{N}\setminus\{0\}$ and every $\alpha>0$ there exists a constant $C_\ell=C_\ell(N, p,\Omega,\alpha)>0$ such that
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha\,|x|}\,,\qquad \text{for every}\ x\in\Omega.
\]
\end{itemize}

\begin{defi}
\label{defi:scp}
Let $V\in L^1_{\rm loc}(\mathbb{R}^N)$ be a non-negative function. We say that $V$ is a {\it strongly confining potential} if it has the following property: there exists an increasing function $\alpha:(0,+\infty)\to(0,+\infty)$ such that
\[
\lim_{R\to +\infty} \alpha(R)=+\infty,
\]
and for every $R>0$ we have
\[
V(x)\ge \alpha(R),\qquad \text{for a.e.}\ x\in \mathbb{R}^N\setminus B_R.
\]
\end{defi}

\begin{coro}
\label{coro:gradienti}
Let $1<p<\infty$ and let $\Omega\subseteq\mathbb{R}^N$ be an open set such that $\lambda_{1,p}(\Omega)>0$. For a constant $\lambda$ such that
\[
\lambda_{1,p}(\Omega)\le \lambda<\mathcal E_p(\Omega),
\]
we consider $v\in W^{1,p}_0(\Omega)$ a nonnegative subsolution of the equation \eqref{subsol}.
Then\footnote{We use the notation $\mathcal{C}$ for the same constant of Proposition \ref{prop:lambdadecay}.}:
\begin{itemize}
\item if $\mathcal{E}_p(\Omega)<+\infty$, there exists $M=M(p,\lambda,\Omega)>0$ such that
 \begin{equation}
\label{decaygrad}
\|\nabla v\|_{L^p(\Omega\setminus B_{R+1})}\le  M\, e^{-\alpha\,R}\,\|v\|_{L^p(\Omega)}, \qquad \text{for every}\ R>0,
\end{equation}
where $\alpha$ is the same exponent as in Proposition \ref{prop:lambdadecay}. Moreover,  for every $0<\beta<\alpha$ there exists a constant
\[
\widetilde{\mathcal{C}}= \widetilde{\mathcal{C}}(\mathcal{C}, p, \lambda, \beta, \alpha-\beta)>0,
\]
such that
\begin{equation}
\label{decaygradiente}
\int_{\Omega}\left|\nabla v\right|^{p}\,e^{\beta p|x|}\,dx\le  \widetilde{\mathcal{C}}\,\|v\|^p_{L^{p}(\Omega)};
\end{equation}
\item if $\mathcal{E}_p(\Omega)=+\infty$, for every $\alpha>0$ there exists a constant $M=M(p, \lambda, \Omega,\alpha)>0$ such that \eqref{decaygrad} holds. Moreover,  for every $\beta>0$ there exists a constant
\[
\widetilde{\mathcal{C}}= \widetilde{\mathcal{C}}(\mathcal{C}, p, \lambda, \beta)>0,
\]
such that \eqref{decaygradiente} holds.
\end{itemize}
In particular, in both cases we have that
\[
\int_\Omega |\nabla v|^\gamma\,dx<+\infty,\qquad \text{for every}\ 0<\gamma\le p.
\]
\end{coro}

\begin{lm}
\label{lm:delcanto}
Let $1<p<\infty$ and let $\Omega\subseteq\mathbb{R}^N$ be an open set. Then
\[
W^{1,p}_0(\Omega)\hookrightarrow L^p(\Omega)\ \text{is compact}\qquad \Longleftrightarrow\qquad \mathcal{E}_p(\Omega)=+\infty.
\]
\end{lm}

\begin{prop}[Exponential decay at infinity]
\label{prop:lambdadecay}
Let $1<p<\infty$ and let $\Omega\subseteq\mathbb{R}^N$ be an open set such that $\lambda_{1,p}(\Omega)>0$. For a constant $\lambda$ such that
\[
\lambda_{1,p}(\Omega)\le \lambda<\mathcal E_p(\Omega),
\]
we consider $v\in W^{1,p}_0(\Omega)$ a nonnegative subsolution of the equation
\begin{equation}\label{subsol}
-\Delta_p u = \lambda\, u^{p-1},\qquad \text{in}\ \Omega.
\end{equation}
Then:
\begin{itemize}
\item if $\mathcal{E}_p(\Omega)<+\infty$,  there exist two constants $\mathcal{C}=\mathcal{C}(N, p, \lambda, \Omega)>0$ and $\alpha=\alpha(p, \lambda, \Omega)>0$ such that  $v$
satisfies the following decay estimate
\begin{equation}
\label{decayinfty}
0\le v(x)\le \mathcal{C}\,\|v\|_{L^p(\Omega)}\,e^{-\alpha\,|x|},\qquad \text{for a.\,e.}\ x\in\Omega.
\end{equation}
Moreover, the constant $\alpha$ decreasingly depends on $\lambda$ and we have
\[
\lim_{\lambda\nearrow \mathcal{E}_p(\Omega)}\alpha=0;
\]
\item if $\mathcal{E}_p(\Omega)=+\infty$, for every $\alpha>0$ there exists a constant $\mathcal{C}=\mathcal{C}(N, p, \lambda, \Omega,\alpha)>0$ such that \eqref{decayinfty} holds.
\end{itemize}
\end{prop}

\begin{equation}
\label{krasnodar}
\gamma(A;X)=\inf\left\{k\in\mathbb{N}\, :\, \exists\ \text{a continuous odd map}\ f:A\to\mathbb{S}^{k-1}\right\},
\end{equation}

\begin{lm}
\label{lm:giggi}
Let $1<p<\infty$ and let $\Omega\subseteq\mathbb{R}^N$ be an open set such that $\lambda_{1,p}(\Omega)>0$. Let $X(\Omega)$ be a Banach space with the following properties:
\begin{itemize}
\item $C^\infty_0(\Omega)\subseteq X(\Omega)$;
\vskip.2cm
\item $X(\Omega)\subseteq W^{1,p}_0(\Omega)$ with continuous inclusion, i.e. there exists $C>0$ such that
\[
\|\nabla \varphi\|_{L^p(\Omega)}\le C\, \|\varphi\|_{X(\Omega)},\qquad \text{for every}\ \varphi\in X(\Omega).
\]
\end{itemize}
We set
\[
\mathcal{X}_{k,p}(\Omega)=\Big\{K\subseteq \mathcal{S}_p(\Omega)\cap X(\Omega)\, :\, K\ \text{symmetric and compact},\, \gamma(K;X(\Omega))\ge k\Big\},
\]
and
\[
\mathcal{W}_{k,p}(\Omega)=\Big\{K\subseteq \mathcal{S}_p(\Omega)\cap W^{1,p}_0(\Omega)\, :\, K\ \text{symmetric and compact},\, \gamma(K;W^{1,p}_0(\Omega))\ge k\Big\}.
\]
Then, for every $K\in \mathcal{W}_{k,p}(\Omega)$ and every $\delta>0$, there exists $K_\delta\in \mathcal{X}_{k,p}(\Omega)$ such that
\[
\max_{\varphi\in K_\delta} \int_\Omega |\nabla \varphi|^p\,dx\le \max_{\varphi\in K} \int_\Omega |\nabla \varphi|^p\,dx+\delta.
\]
\end{lm}

\subsection{Beneath the essential spectrum}
Let us start by considering the spectrum of the Dirichlet-Laplacian on an open set $\Omega\subseteq\mathbb{R}^N$. Whenever the embedding
\begin{equation}
\label{mimmergo}
W^{1,2}_0(\Omega)\hookrightarrow L^2(\Omega),
\end{equation}
is compact, we know that this spectrum is discrete, made of a diverging sequence of positive eigenvalues $\{\lambda_k(\Omega)\}_{k\in\mathbb{N}\setminus\{0\}}$, each one having a finite multiplicity. Characterizations of open sets for which the embedding \eqref{mimmergo} is compact can be found for example in \cite[Chapter 15]{Maz}.
\par
Moreover, in this situation we can build a system of orthonormal (in $L^2$ sense) associated eigenfunctions $\{u_k\}_{k\in\mathbb{N}\setminus\{0\}}\subseteq W^{1,2}_0(\Omega)$, which gives a Hilbertian basis of $L^2(\Omega)$. Accordingly, we can characterize the eigenvalues as follows
\[
\lambda_1(\Omega)=\inf_{\varphi\in W^{1,2}_0(\Omega)} \left\{\int_\Omega |\nabla\varphi|^2\,dx\, :\, \int_\Omega |\varphi|^2\,dx=1\right\},
\]
and for $k\ge 2$
\[
\lambda_k(\Omega)=\inf_{\varphi\in W^{1,2}_0(\Omega)} \left\{\int_\Omega |\nabla\varphi|^2\,dx\, :\, \int_\Omega |\varphi|^2\,dx=1,\ \int_\Omega \varphi\,u_j\,dx=0, \ \text{for}\ j=1,\dots,k-1\right\}.
\]
Each eigenfunction $u_k$ is a minimizer of the corresponding problem.
See for example \cite[Chapter VI]{CH} or \cite[Chapter 4, Section 8]{BraBook} for these facts.
\par
We also recall that an alternative characterization of the eigenvalues is given by the well-known {\it Courant-Fischer formula}
\begin{equation}
\label{courant-fischer}
\lambda_k(\Omega)=\inf\left\{ \max_{\varphi\in K\cap \mathcal{S}_2(\Omega)} \int_\Omega|\nabla \varphi|^2\,dx\,:\, K \subseteq W^{1,2}_0(\Omega)\ \mbox{subspace with}\ \dim K=k \right\},
\end{equation}
where
\[
\mathcal{S}_2(\Omega):=\Big\{u\in L^2(\Omega)\, :\, \|u\|_{L^2(\Omega)}=1\Big\}.
\]
However, for a generic open set, it may happen that the embedding \eqref{mimmergo} fails to be compact. In this case, things become more complicated and intriguing: the spectrum of the Dirichlet-Laplacian stops being purely discrete (see \cite[Theorem 10.1.5]{BS}) and we have a non-empty {\it essential spectrum}, indicated by $\sigma_{\rm ess}(\Omega)$.
\par
We recall that the latter can be characterized in terms of {\it singular constrained Palais-Smale sequences} (see for example \cite[Theorem 9.1.2]{BS} or \cite[Theorem 7.2]{HS}). Namely, we have that
$\lambda\in \sigma_{\rm ess}(\Omega)$ if and only if there exists a sequence $\{\varphi_n\}_{n\in\mathbb{N}}\subseteq W^{1,2}_0(\Omega)$ such that
\begin{enumerate}
\item $\varphi_n\in\mathcal{S}_2(\Omega)$, for every $n\in\mathbb{N}$;
\vskip.2cm
\item $\lim\limits_{n\to\infty}\|\nabla \varphi_n\|_{L^2(\Omega)}^2=\lambda$;
\vskip.2cm
\item we have
\[
\lim_{n\to\infty} \|-\Delta \varphi_n-\lambda\,\varphi_n\|_{W^{-1,2}(\Omega)}=0;
\]
\item $\{\varphi_n\}_{n\in\mathbb{N}}$ weakly converges to $0$ in $L^2(\Omega)$.
\end{enumerate}
A celebrated result by Persson (see \cite[Theorem 2.1]{Pe}) gives a characterization of ``geometric''  flavour to the bottom of $\sigma_{\rm ess}(\Omega)$.
More precisely, if we introduce the {\it Poincar\'e constant ``at infinity''}
\[
\mathcal{E}(\Omega):=\sup_{R>0}\lambda_1\big(\Omega\setminus \overline{B_R}\big),
\]
then Persson's Theorem asserts that
\begin{equation}
\label{perso}
\inf \Big\{\lambda>0\, :\, \lambda\in\sigma_{\rm ess}(\Omega)\Big\}=\mathcal{E}(\Omega).
\end{equation}
Nevertheless, even when $\sigma_{\rm ess}(\Omega)\not=\emptyset$, it may happen that some discrete eigenvalues with finite multiplicities ``pop-up'' beneath the essential spectrum, i.e. there may exist some eigenvalues $\lambda<\mathcal{E}(\Omega)$. In this case, we have that these eigenvalues can still be characterized through the variational principles recalled above (see for example \cite[Theorems 10.2.1 \& 10.2.2]{BS}).

\par
We also recall that the corresponding eigenfunctions are strongly localized in space, in the sense that they exponentially ``fall-off'' at infinity.  In other words, they enjoy an estimate of the form
\begin{equation}
\label{falloff}
|u(x)|\le C\,e^{-\alpha\,|x|},\qquad \text{for}\ x\in\Omega,
\end{equation}
for suitable $C,\alpha>0$. One of the very first papers in the literature proving this kind of results is \cite{Sn} by \v{S}nol, for Schr\"odinger operators of the form $-\Delta+V$, with a non-negative potential $V$. We also wish to cite two papers \cite{Si, Si2} by Simon, which are connected with our main result presented below.
\par
Nowadays, a result of the type \eqref{falloff} is a very particular instance of decay estimates that are usually named after Agmon, following his influential work \cite{Ag}.
There, one can find a systematic study of decay estimates for eigenfunctions of $-\Delta+V$. Since then, it has become a classical subject in Spectral Theory. Many authors have contributed, by proving extensions, generalizations and refinements. Without any attempt of completeness, we mention the papers \cite{AH2OM} and \cite{CS}. An overview on Agmon estimates and their applications can be found in \cite{He1,He2}.

\subsection{Eigenvalues of the $p-$Laplacian}
The aim of this paper is to extend to the case of the Dirichlet $p-$Laplacian the previous analysis of
\[
``\text{\it eigenvalues lying beneath the essential spectrum\,}''.
\]
This sentence requires some precisions: at first, what it is intended by {\it eigenvalue} of the Dirichlet $p-$Laplacian on an open set $\Omega\subseteq\mathbb{R}^N$. From the point of view of Critical Point Theory, this is readily said: these eigenvalues can be understood as critical values of the functional
\[
\varphi\mapsto\int_\Omega |\nabla \varphi|^p\,dx,
\]
constrained to the ``sphere''
\[
\mathcal{S}_p(\Omega)=\Big\{u\in L^p(\Omega)\, :\, \|u\|_{L^p(\Omega)}=1\Big\}.
\]
The associated critical points are the eigenfunctions, accordingly.
Thus, in other words, eigenvalues of the Dirichlet $p-$Laplacian are those numbers $\lambda$ such that there exists $u\in W^{1,p}_0(\Omega)\setminus\{0\}$ weakly solving
\[
-\Delta_p u=\lambda\,|u|^{p-2}\,u,\qquad \text{in}\ \Omega,
\]
that is
\[
\int_\Omega \langle |\nabla u|^{p-2}\,\nabla u,\nabla\varphi\rangle\,dx=\lambda\,\int_\Omega |u|^{p-2}\,u\,\varphi,\qquad \text{for every}\ \varphi\in C^\infty_0(\Omega).
\]
Observe that the global infimum of the constrained Dirichlet integral
\[
\lambda_{1,p}(\Omega):=\inf_{\varphi\in W^{1,p}_0(\Omega)} \left\{\int_\Omega |\nabla\varphi|^p\,dx\, :\, \int_\Omega |\varphi|^p\,dx=1\right\},
\]
corresponds to the {\it first eigenvalue} or {\it ground state energy}.
\par
On the other hand, what is intended by {\it essential spectrum} in this context is quite unclear. Nevertheless, by recalling Persson's result \eqref{perso} for the linear case, one might bravely guess that the {\it $L^p$ Poincar\'e constant at infinity}
\[
\mathcal{E}_{p}(\Omega):=\sup_{R>0}\lambda_{1,p}\big(\Omega\setminus \overline{B_R}\big),
\]
could be a valid surrogate for the infimum of the essential spectrum, in this nonlinear context. We are then lead to the following questions: is it possible to produce eigenvalues of the Dirichlet $p-$Laplacian below the threshold $\mathcal{E}_p(\Omega)$? Do the corresponding eigenfunctions enjoy a decay estimate similar to \eqref{falloff}? Attempting  to answer these questions, we will try to take very minimal assumptions on the open set $\Omega$.
\vskip.2cm\noindent
Let us start with the simpler case $\mathcal{E}_p(\Omega)=+\infty$: in this case, we have that the embedding
\begin{equation}
\label{mimmergop}
W^{1,p}_0(\Omega)\hookrightarrow L^p(\Omega),
\end{equation}
is compact, see for example Lemma \ref{lm:delcanto} below. In this scenario, it is well-known that we can produce a diverging sequence of positive eigenvalues by suitably mimicking the minmax Courant-Fischer procedure. This situation is very much studied in the literature, we just mention few classical references \cite{GP, Le, Szu}, the lecture notes \cite{Lin} and the PhD Thesis \cite{Fra}, where one can find many more references.
\par
We briefly recall the construction of this sequence of eigenvalues:
for every $k\in\mathbb{N}\setminus\{0\}$, we set
\[
\mathcal{W}_{k,p}(\Omega)=\Big\{K\subseteq \mathcal{S}_p(\Omega)\cap W^{1,p}_0(\Omega)\, :\, K\ \text{symmetric and compact},\, \gamma(K;W^{1,p}_0(\Omega))\ge k\Big\}.
\]
Here $\gamma(K;W^{1,p}_0(\Omega))$ indicates the {\it Krasnosel'ski\u{\i} genus} of $K$ in the norm topology of $W^{1,p}_0(\Omega)$, see \eqref{krasnodar} below for the definition.
With this class of compact sets at hand, we can define the {\it $k-$th minmax Ljusternik-Schnirelmann value} as follows
\begin{equation}
\label{minmaxLS}
\lambda_{k,p}^{\rm LS}(\Omega):=\inf_{K\in \mathcal{W}_{k,p}(\Omega)} \max_{\varphi\in K} \int_\Omega |\nabla \varphi|^p\,dx,\qquad k\in\mathbb{N}\setminus\{0\},
\end{equation}
and prove that this actually defines an eigenvalue, if the embedding \eqref{mimmergop} is compact. Let us notice that, for $k=1$, the previous definition boils down $\lambda_{1,p}(\Omega)$, i.e. it really gives the first eigenvalue. For this reason, in what follows we will omit the superscript ``${\rm LS}$'' in the case $k=1$. Moreover,  it is also possible to prove that, for  $k=2$, on a connected open set we have
\[
\lambda_{2,p}^{\rm LS}(\Omega)=\min\Big\{\lambda>\lambda_{1,p}(\Omega)\,:\, \lambda\ \text{is an eigenvalue}\Big\},
\]
see for instance \cite[Theorem 3.4]{JL}.
\begin{rem}
In order to appreciate these apparently pedantic remarks about $k=1$ and $k=2$, the reader should keep in mind that it is still a major open problem to know whether $\{\lambda_{k,p}^{\rm LS}(\Omega)\}_{n\in\mathbb{N}\setminus\{0\}}$ exhaust the whole spectrum or not, at least in the case when \eqref{mimmergop} is compact. We also recall that it would be possible
to define the previous minmax levels by means of another index $i$, in place of the Krasnosel'ski\u{\i} genus: for example, one could use the {\it $\mathbb{Z}_2-$cohomological index}
\cite{fadell_rabinowitz1977} or the {\it Ljusternik-Schnirelman Category} \cite[Chapter 2]{Str}.
\end{rem}
We then consider the case $\mathcal{E}_p(\Omega)<+\infty$: here the embedding \eqref{mimmergop} fails to be compact, see again Lemma \ref{lm:delcanto} below. Nevertheless, one could still define the minmax values \eqref{minmaxLS} and reasonably argue that whenever
\[
\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega),
\]
we still have an eigenvalue, similarly to what happens in the linear case...
\subsection{Main result}
We will show that this is actually the case. Moreover, we will also get a \v{S}nol-Simon--type decay estimate for the relevant eigenfunction (we refer to Remark \ref{rem:SS} for the terminology). Namely, we prove the following

\subsection{Eigenvalues of the $p-$Laplacian}
The aim of this paper is to extend to the case of the Dirichlet $p-$Laplacian the previous analysis of
\[
``\text{\it eigenvalues lying beneath the essential spectrum\,}''.
\]
This sentence requires some precisions: at first, what it is intended by {\it eigenvalue} of the Dirichlet $p-$Laplacian on an open set $\Omega\subseteq\mathbb{R}^N$. From the point of view of Critical Point Theory, this is readily said: these eigenvalues can be understood as critical values of the functional
\[
\varphi\mapsto\int_\Omega |\nabla \varphi|^p\,dx,
\]
constrained to the ``sphere''
\[
\mathcal{S}_p(\Omega)=\Big\{u\in L^p(\Omega)\, :\, \|u\|_{L^p(\Omega)}=1\Big\}.
\]
The associated critical points are the eigenfunctions, accordingly.
Thus, in other words, eigenvalues of the Dirichlet $p-$Laplacian are those numbers $\lambda$ such that there exists $u\in W^{1,p}_0(\Omega)\setminus\{0\}$ weakly solving
\[
-\Delta_p u=\lambda\,|u|^{p-2}\,u,\qquad \text{in}\ \Omega,
\]
that is
\[
\int_\Omega \langle |\nabla u|^{p-2}\,\nabla u,\nabla\varphi\rangle\,dx=\lambda\,\int_\Omega |u|^{p-2}\,u\,\varphi,\qquad \text{for every}\ \varphi\in C^\infty_0(\Omega).
\]
Observe that the global infimum of the constrained Dirichlet integral
\[
\lambda_{1,p}(\Omega):=\inf_{\varphi\in W^{1,p}_0(\Omega)} \left\{\int_\Omega |\nabla\varphi|^p\,dx\, :\, \int_\Omega |\varphi|^p\,dx=1\right\},
\]
corresponds to the {\it first eigenvalue} or {\it ground state energy}.
\par
On the other hand, what is intended by {\it essential spectrum} in this context is quite unclear. Nevertheless, by recalling Persson's result \eqref{perso} for the linear case, one might bravely guess that the {\it $L^p$ Poincar\'e constant at infinity}
\[
\mathcal{E}_{p}(\Omega):=\sup_{R>0}\lambda_{1,p}\big(\Omega\setminus \overline{B_R}\big),
\]
could be a valid surrogate for the infimum of the essential spectrum, in this nonlinear context. We are then lead to the following questions: is it possible to produce eigenvalues of the Dirichlet $p-$Laplacian below the threshold $\mathcal{E}_p(\Omega)$? Do the corresponding eigenfunctions enjoy a decay estimate similar to \eqref{falloff}? Attempting to answer these questions, we will try to take very minimal assumptions on the open set $\Omega$.
\vskip.2cm\noindent
Let us start with the simpler case $\mathcal{E}_p(\Omega)=+\infty$: in this case, we have that the embedding
\begin{equation}
\label{mimmergop}
W^{1,p}_0(\Omega)\hookrightarrow L^p(\Omega),
\end{equation}
is compact, see for example Lemma \ref{lm:delcanto} below. In this scenario, it is well-known that we can produce a diverging sequence of positive eigenvalues by suitably mimicking the minmax Courant-Fischer procedure. This situation is very much studied in the literature, we just mention few classical references \cite{GP, Le, Szu}, the lecture notes \cite{Lin} and the PhD Thesis \cite{Fra}, where one can find many more references.
\par
We briefly recall the construction of this sequence of eigenvalues:
for every $k\in\mathbb{N}\setminus\{0\}$, we set
\[
\mathcal{W}_{k,p}(\Omega)=\Big\{K\subseteq \mathcal{S}_p(\Omega)\cap W^{1,p}_0(\Omega)\, :\, K\ \text{symmetric and compact},\, \gamma(K;W^{1,p}_0(\Omega))\ge k\Big\}.
\]
Here $\gamma(K;W^{1,p}_0(\Omega))$ indicates the {\it Krasnosel'ski\u{\i} genus} of $K$ in the norm topology of $W^{1,p}_0(\Omega)$, see \eqref{krasnodar} below for the definition.
With this class of compact sets at hand, we can define the {\it $k-$th minmax Ljusternik-Schnirelmann value} as follows
\begin{equation}
\label{minmaxLS}
\lambda_{k,p}^{\rm LS}(\Omega):=\inf_{K\in \mathcal{W}_{k,p}(\Omega)} \max_{\varphi\in K} \int_\Omega |\nabla \varphi|^p\,dx,\qquad k\in\mathbb{N}\setminus\{0\},
\end{equation}
and prove that this actually defines an eigenvalue, if the embedding \eqref{mimmergop} is compact. Let us notice that, for $k=1$, the previous definition boils down $\lambda_{1,p}(\Omega)$, i.e. it really gives the first eigenvalue. For this reason, in what follows we will omit the superscript ``${\rm LS}$'' in the case $k=1$. Moreover, it is also possible to prove that, for $k=2$, on a connected open set we have
\[
\lambda_{2,p}^{\rm LS}(\Omega)=\min\Big\{\lambda>\lambda_{1,p}(\Omega)\,:\, \lambda\ \text{is an eigenvalue}\Big\},
\]
see for instance \cite[Theorem 3.4]{JL}.
\begin{rem}
In order to appreciate these apparently pedantic remarks about $k=1$ and $k=2$, the reader should keep in mind that it is still a major open problem to know whether $\{\lambda_{k,p}^{\rm LS}(\Omega)\}_{n\in\mathbb{N}\setminus\{0\}}$ exhaust the whole spectrum or not, at least in the case when \eqref{mimmergop} is compact. We also recall that it would be possible
to define the previous minmax levels by means of another index $i$, in place of the Krasnosel'ski\u{\i} genus: for example, one could use the {\it $\mathbb{Z}_2-$cohomological index}
\cite{fadell_rabinowitz1977} or the {\it Ljusternik-Schnirelman Category} \cite[Chapter 2]{Str}.
\end{rem}
We then consider the case $\mathcal{E}_p(\Omega)<+\infty$: here the embedding \eqref{mimmergop} fails to be compact, see again Lemma \ref{lm:delcanto} below. Nevertheless, one could still define the minmax values \eqref{minmaxLS} and reasonably argue that whenever
\[
\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega),
\]
we still have an eigenvalue, similarly to what happens in the linear case...
\subsection{Main result}
We will show that this is actually the case. Moreover, we will also get a \v{S}nol-Simon--type decay estimate for the relevant eigenfunction (we refer to Remark \ref{rem:SS} for the terminology). Namely, we prove the following

\par
We also recall that the corresponding eigenfunctions are strongly localized in space, in the sense that they exponentially ``fall-off'' at infinity. In other words, they enjoy an estimate of the form
\begin{equation}
\label{falloff}
|u(x)|\le C\,e^{-\alpha\,|x|},\qquad \text{for}\ x\in\Omega,
\end{equation}
for suitable $C,\alpha>0$. One of the very first papers in the literature proving this kind of results is \cite{Sn} by \v{S}nol, for Schr\"odinger operators of the form $-\Delta+V$, with a non-negative potential $V$. We also wish to cite two papers \cite{Si, Si2} by Simon, which are connected with our main result presented below.
\par
Nowadays, a result of the type \eqref{falloff} is a very particular instance of decay estimates that are usually named after Agmon, following his influential work \cite{Ag}.
There, one can find a systematic study of decay estimates for eigenfunctions of $-\Delta+V$. Since then, it has become a classical subject in Spectral Theory. Many authors have contributed, by proving extensions, generalizations and refinements. Without any attempt of completeness, we mention the papers \cite{AH2OM} and \cite{CS}. An overview on Agmon estimates and their applications can be found in \cite{He1,He2}.

\subsection{Plan of the paper}
The paper starts with some preliminary materials contained in Section \ref{sec:2}. The main content of Section \ref{sec:3} is the \v{S}nol-Simon--type estimate of Proposition \ref{prop:lambdadecay}, which holds more generally for non-negative {\it subsolutions} of the eigenvalue equation. As a consequence, we also obtain some decay and integrability properties for their gradients. Section \ref{sec:4} is devoted to the ``perturbed'' eigenvalue problem described above, obtained by adding a strongly confining potential. Here, the technical Lemma \ref{lm:giggi} may be of some independent interest. The proof of the Main Theorem is contained in Section \ref{sec:5}, while in Section \ref{sec:6} we provide some examples of unbounded open sets to which our result applies. The paper ends with two appendices: in particular, in Appendix \ref{sec:B} we show that, for $p=2$, the minmax Ljusternik-Schnirelmann values below the essential spectrum coincide with the eigenvalues given by the Courant-Fischer formula.

\begin{prop}
\label{prop:tabaccaio}
Let $\Omega\subseteq\mathbb{R}^N$ be an open set and let us suppose that there exists $k\in\mathbb{N}\setminus\{0\}$ such that
\begin{equation}\
\label{PLC}
\lambda_{k,2}^{\rm LS}(\Omega)<\mathcal{E}(\Omega):=\sup_{R>0} \lambda_1(\Omega\setminus \overline{B_R}).
\end{equation}
Then, for every $\ell\in\{1,\dots,k\}$ we have
\[
\lambda_{\ell,2}^{\rm LS}(\Omega)=\lambda_{\ell}(\Omega).
\]
\end{prop}
\begin{proof}
We can limit ourselves to the case $\mathcal{E}(\Omega)<+\infty$, otherwise the result is already contained in \cite[Theorem A.2]{BraParSqu}.
We may suppose that $k\ge 2$, otherwise the claim is straightforward. We start by observing that for every vector subspace $K\subseteq W^{1,2}_0(\Omega)$ having dimension $\ell$, it holds
\[
K\cap\mathcal{S}_2(\Omega)\in \mathcal{W}_{\ell,2}(\Omega),
\]
i.e. $K\cap\mathcal{S}_2(\Omega)$ is a compact symmetric subset of $W^{1,2}_0(\Omega)$, with genus equal to $\ell$ (see for example \cite[Chapter II, Proposition 5.2]{Str}). This entails that
\[
\lambda_{\ell,2}^{\rm LS}(\Omega)\le \lambda_\ell(\Omega).
\]
In order to prove the reverse inequality, we will use an approximation argument\footnote{This route will avoid considering the Spectral Resolution of the Dirichlet-Laplacian on $\Omega$ (see for example \cite[Chapter 6]{BS}). Indeed, this last operator could have a continuous part in its spectrum, under the standing assumptions on the open set. On the contrary, the approximation argument will only need the spectral properties of operators having a discrete spectrum. The reader with a sufficiently good expertise in Spectral Theory will certainly find our argument a bit akward.}. Namely, let us consider the same operators of Proposition \ref{prop:approssimazione}, i.e.
\[
\varphi\mapsto-\Delta\varphi+V_n\,\varphi,
\]
with homogeneous Dirichlet boundary conditions. The quadratic form naturally associated to this operator is given by the functional $\mathcal{G}_n$, defined in Proposition \ref{prop:approssimazione}. By classical Spectral Theory, for every fixed $n\in\mathbb{N}\setminus\{0\}$, the compactness of the embedding $W^{1,2}_0(\Omega;V)\hookrightarrow L^2(\Omega)$ guarantees that such an operator has a discrete spectrum, made of a diverging sequence of positive eigenvalues, each one with finite multiplicity. They can be characterized through the Courant-Fischer formula
\[
\lambda_{\ell}(\Omega;V_n):=\inf\left\{ \max_{u\in K\cap \mathcal{S}_2(\Omega)} \mathcal{G}_n(u)\,:\, K \subseteq W^{1,2}_0(\Omega;V)\ \mbox{subspace with}\ \dim K=\ell \right\}.
\]
Moreover, the infimum on the right-hand side is attained by
\[
K_\ell=\mathrm{Vect}\Big(\left\{u_{1,n},\dots,u_{\ell,n}\right\}\Big),
\]
i.e. by the vector subspace generated by the first $\ell$ eigenfunctions of the operator. In fact, these eigenfunctions can be chosen so that
\[
\int_\Omega u_{i,n}\,u_{j,n}\,dx=\delta_{ij},\qquad \text{for every}\ n\in\mathbb{N},\ i,j\in\{1,\dots,\ell\},
\]
and the following characterization also holds:
\[
\lambda_{\ell}(\Omega;V_n)=\min_{\varphi\in \mathcal{S}_2(\Omega)\cap W^{1,2}_0(\Omega;V)}\left\{\mathcal{G}_n(\varphi)\,:\, \int_\Omega \varphi\,u_{i,n}\,dx=0,\ i\in\{1,\dots,\ell-1\}\right\}.
\]
We are now ready to conclude the proof: according to \cite[Theorem A.2]{BraParSqu}, for every $\ell\in \mathbb{N}\setminus\{0\}$ we have
\[
\lambda_{\ell,2}^{\rm LS}(\Omega;V_n)=\lambda_{\ell}(\Omega;V_n)=\int_\Omega |\nabla u_{\ell,n}|^2\,dx+\int_\Omega V_n\,|u_{\ell,n}|^2\,dx,
\]
i.e. the equality claimed in the statement holds true for the approximating operator.
By the proof of the Main Theorem, we know that each $\{u_{i,n}\}_{n\in\mathbb{N}}$ converges strongly in $L^2(\Omega)$ to an eigenfunction $u_i$ of the Dirichlet-Laplacian on $\Omega$, up to a subsequence. In particular, we still have
\begin{equation}
\label{ortaggi}
\int_\Omega u_{i}\,u_{j}\,dx=\delta_{ij},\qquad \text{for every}\ i,j\in\{1,\dots,\ell\}.
\end{equation}
By virtue of Proposition \ref{prop:approssimazione} and by the lower semicontinuity of the Dirichlet integral, we thus get for every $\ell\in\{1,\dots,k\}$
\[
\begin{split}
\lambda_{\ell,2}^{\rm LS}(\Omega)=\lim_{n\to\infty} \lambda_{\ell,2}^{\rm LS}(\Omega;V_n)\ge\lim_{n\to\infty} \int_\Omega |\nabla u_{\ell,n}|^2\,dx\ge \int_\Omega |\nabla u_\ell|^2\,dx\\
\end{split}
\]
On the other hand, by defining the $\ell-$dimensional vector subspace
\[
\overline{K}=\mathrm{Vect}\Big(\left\{u_{1},\dots,u_{\ell}\right\}\Big),
\]
we get
\[
\begin{split}
\lambda_\ell(\Omega)\le \max_{\varphi\in \overline{K}\cap\mathcal{S}_2(\Omega)} \int_\Omega |\nabla\varphi|^2\,dx&=\max_{\alpha=(\alpha_1,\dots,\alpha_\ell)\in \mathbb{S}^{\ell-1}} \int_\Omega \left|\sum_{i=1}^\ell \alpha_i\,\nabla u_i\right|^2\,dx\\
&=\max_{\alpha=(\alpha_1,\dots,\alpha_\ell)\in \mathbb{S}^{\ell-1}} \sum_{i=1}^\ell |\alpha_i|^2\,\int_\Omega |\nabla u_i|^2\,dx\le \int_\Omega |\nabla u_\ell|^2\,dx,
\end{split}
\]
thanks to the orthogonality relations \eqref{ortaggi} and the fact that each $u_i$ is an eigenfunction. In turn, we get the reverse estimate
\[
\lambda_{\ell,2}^{\rm LS}(\Omega)\ge \lambda_\ell(\Omega),
\]
as desired.
\end{proof}

The rest of this subsection is devoted to a couple of (detailed) comments on this result. We start with the exponential fall-off.
\begin{rem}[Fall-off]
\label{rem:SS}
As for the ``compact case'' $\mathcal{E}_p(\Omega)=+\infty$, our decay estimate represents an extension to the nonlinear case of \cite[Theorem 2]{Si2} by Simon, recalled above. The latter deals with Schr\"odinger operators $-\Delta+V$ on the whole $\mathbb{R}^N$, under the effect of a non-negative strongly confining potential $V$ (see Definition \ref{defi:scp} below).
\par
In the ``non-compact case'' $\mathcal{E}_p(\Omega)<+\infty$, our estimate is a nonlinear variation on the classical result by \v{S}nol, previously recalled. In particular, we have that
\[
\alpha_k\le \dots\le \alpha_1,
\]
with the notation above.
That is, the higher the eigenvalue, the slower the exponential decay. Moreover, these exponents deteriorate (i.e. they approach $0$) as the eigenvalues approach the spectral threshold $\mathcal{E}_p(\Omega)$: we refer to Proposition \ref{prop:lambdadecay} below, for this fact.
This behaviour is in perfect accordance with \v{S}nol's result.
Indeed, in \cite[Theorem pag. 282]{Sn}, this estimate is proved for eigenfunctions of $-\Delta+V$, corresponding to an eigenvalue $\lambda$ below of the essential spectrum $\mathcal{E}_V$. Also in \cite{Sn}, the exponent $\alpha$ dictating the fall-off depends on the difference $\mathcal{E}_V-\lambda$ and it goes to $0$, as this difference goes to $0$. More precisely, \v{S}nol's exponent is given by
\[
\alpha=\log\left(1+\frac{\mathcal{E}_V-\lambda}{K}\right),
\]
for constant $K$ depending on the (positive) potential $V$.
 For these reasons, in what follows we will refer to the estimate of the Main Theorem as {\it \v{S}nol-Simon--type decay estimate}.
\par
Finally, we wish to point out that our method of proof is different from those of both \cite{Si2} and \cite{Sn} and based on nonlinear techniques: we first obtain exponential decay in $L^p$ norm and then use an $L^\infty-L^p$ estimate ``localized at infinity'' to transform it into a pointwise decay estimate. The decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\'e inequalities, while the second estimate is a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.
\par
It is worth mentioning that, once we obtain decay estimates on the eigenfunctions, it is possible to exploit the equation so to obtain some decay and integrability estimates on their gradients, as well (see Corollary \ref{coro:gradienti} below).
\end{rem}
We then briefly comment on how we prove the existence part of the Main Theorem.
\begin{rem}[Existence of eigenfunctions]
There are various ways to get existence of eigenfunctions. Of course, we can focus on the case $\mathcal{E}_p(\Omega)<+\infty$, in light of what we have previously recalled. In the Hilbertian setting (i.\,e. for $p=2$), a purely variational approach based on concentration-compactness arguments is provided by Smets, see \cite[Theorem 4.1 $\&$ Theorem 4.2]{Sm}, for a slightly different eigenvalue problem. His approach relies on the possibility to characterize higher eigenvalues as an infimum, possibly on the orthogonal complement of a finite dimensional vector space. This fact is true for $p=2$, but it has no counterpart for $p \neq 2$. For this reason, it is less clear whether this approach can be applied for the full range of $p$, see for instance \cite[Remark 4.4 (b)]{Sm}.
\par
We briefly explain our route. We use a {\it spectral stability argument}, i.e. we prove that our minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.
More precisely, we ``compactify'' the original eigenvalue problem by considering a new family of problems, each one obtained by adding a strongly confining potential $V_n$. For this family, we can prove existence of infinitely many ``perturbed'' eigenvalues, by the usual minmax procedure recalled above. Moreover, if $V_n$ tends to $0$ (in a suitable sense) as $n$ goes to $\infty$, we can prove that these eigenvalues converge to the original minmax Ljusternik-Schnirelmann values. Then, in order to prove that those lying below $\mathcal{E}_p(\Omega)$ actually define an eigenvalue, we aim to infer strong convergence in $L^p$ of the ``perturbed'' eigenfunctions. Here, we crucially exploit that these ``perturbed'' eigenfunctions enjoy a uniform \v{S}nol-Simon decay estimate, whenever they are associated to a ``perturbed'' eigenvalue which is converging to a $\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega)$. The uniform fall-off in turn implies an $L^p$ equi-tightness property, which permits to appeal to the classical Riesz-Fr\'echet-Kolmogorov compactness criterion. In a nutshell, we obtain compactness through regularity.
\par
In the language of Calculus of Variations, this is essentially a $\Gamma-$convergence--type argument in disguise, but used for higher critical points, other than the global minimum (see \cite{Brai,Dal} for the general theory of $\Gamma-$convergence). In doing this, we crucially borrow some ideas from the paper \cite{CD} by Champion and De Pascale.
\end{rem}

\subsection{Plan of the paper}
The paper starts with some preliminary materials contained in Section \ref{sec:2}. The main content of Section \ref{sec:3} is the \v{S}nol-Simon--type estimate of Proposition \ref{prop:lambdadecay}, which holds more generally for non-negative {\it subsolutions} of the eigenvalue equation. As a consequence, we also obtain some decay and integrability properties for their gradients. Section \ref{sec:4} is devoted to the ``perturbed'' eigenvalue problem described above, obtained by adding a strongly confining potential. Here, the technical Lemma \ref{lm:giggi} may be of some independent interest. The proof of the Main Theorem is contained in Section \ref{sec:5}, while in Section \ref{sec:6} we provide some examples of unbounded open sets to which our result applies. The paper ends with two appendices: in particular, in Appendix \ref{sec:B} we show that, for $p=2$, the minmax Ljusternik-Schnirelmann values below the essential spectrum coincide with the eigenvalues given by the Courant-Fischer formula.

\begin{ack}
We thank Bernard Helffer for some discussions on Agmon--type estimates and for providing us a copy of his papers \cite{He1, He2}.
We also thank Giovanni Franzina for some discussions on nonlinear eigenvalue problems and for a detailed reading of a preliminary draft of this paper.
\par
L.\,Briani and F.\, Prinari are both members of the {\it Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a
e le loro Applicazioni} (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). They both gratefully acknowledge the financial support of the project GNAMPA 2026  ``Problemi di ottimizzazione di forma in contesti anisotropi e non-locali" ({\tt  CUP E53C25002010001}).
\par
L.\,Brasco has been financially supported by the {\it Fondo di Ateneo per la Ricerca} FAR 2024 and the {\it Fondo per l'Incentivazione alla Ricerca Dipartimentale} FIRD 2025 of the University of Ferrara.
The research of L.\,Briani has been supported by the DFG through the Emmy Noether Programme (project number 509436910).
\end{ack}

For the \v{S}nol-Simon--type decay estimate in Theorem~\ref{PL}, the authors stress that their proof is “based on nonlinear techniques”: they “first obtain exponential decay in $L^p$ norm and then use an $L^\infty-L^p$ estimate ‘localized at infinity’ to transform it into a pointwise decay estimate.” The “decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\'e inequalities,” while the second step uses “a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.”

For the existence/eigenvalue part of Theorem~\ref{PL}, they “use a \emph{spectral stability argument},” i.e. “prove that [the] minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.” Concretely, they “\emph{compactify}” the problem by adding “a strongly confining potential $V_n$,” obtaining “infinitely many ‘perturbed’ eigenvalues, by the usual minmax procedure.” If $V_n\to0$ suitably, these perturbed eigenvalues “converge to the original minmax Ljusternik-Schnirelmann values.” To show that limits below $\mathcal{E}_p(\Omega)$ yield actual eigenvalues, they seek “strong convergence in $L^p$ of the ‘perturbed’ eigenfunctions,” using that these enjoy a “uniform \v{S}nol-Simon decay estimate” when the corresponding perturbed eigenvalues converge to some $\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega)$. This “uniform fall-off… implies an $L^p$ equi-tightness property,” allowing the “Riesz-Fr\'echet-Kolmogorov compactness criterion”; “in a nutshell, we obtain compactness through regularity.” They also describe this as “essentially a $\Gamma$-convergence--type argument in disguise, but used for higher critical points.”

For the \v{S}nol-Simon--type decay estimate in Theorem~\ref{PL}, the authors stress that their proof is “based on nonlinear techniques”: they “first obtain exponential decay in $L^p$ norm and then use an $L^\infty-L^p$ estimate ‘localized at infinity’ to transform it into a pointwise decay estimate.” The “decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\'e inequalities,” while the second step uses “a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.”

For the existence/eigenvalue part of the main theorem, they “use a \emph{spectral stability argument},” i.e. “prove that [the] minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.” Concretely, they “\emph{compactify}” the problem by adding “a strongly confining potential $V_n$,” obtaining “infinitely many ‘perturbed’ eigenvalues, by the usual minmax procedure.” If $V_n\to0$ suitably, these perturbed eigenvalues “converge to the original minmax Ljusternik-Schnirelmann values.” To show that limits below $\mathcal{E}_p(\Omega)$ yield actual eigenvalues, they seek “strong convergence in $L^p$ of the ‘perturbed’ eigenfunctions,” using that these enjoy a “uniform \v{S}nol-Simon decay estimate” when the corresponding perturbed eigenvalues converge to some $\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega)$. This “uniform fall-off… implies an $L^p$ equi-tightness property,” allowing the “Riesz-Fr\'echet-Kolmogorov compactness criterion”; “in a nutshell, we obtain compactness through regularity.” They also describe this as “essentially a $\Gamma$-convergence--type argument in disguise, but used for higher critical points.”

Question:
Let \(1<p<\infty\) and let \(\Omega\subseteq\mathbb{R}^N\) be an open set. Define
\[
\mathcal{S}_p(\Omega)=\{u\in L^p(\Omega):\ \|u\|_{L^p(\Omega)}=1\}.
\]
For each \(k\in\mathbb{N}\setminus\{0\}\), let
\[
\lambda_{k,p}^{\rm LS}(\Omega):=\inf_{K\in \mathcal{W}_{k,p}(\Omega)}\ \max_{\varphi\in K}\int_\Omega |\nabla \varphi|^p\,dx,
\]
where \(\mathcal{W}_{k,p}(\Omega)\) is the family of symmetric compact sets \(K\subseteq \mathcal{S}_p(\Omega)\cap W^{1,p}_0(\Omega)\) whose Krasnosel'skii genus is at least \(k\). Also define
\[
\lambda_{1,p}(U):=\inf\left\{\int_U |\nabla \varphi|^p\,dx:\ \varphi\in W^{1,p}_0(U),\ \int_U |\varphi|^p\,dx=1\right\}
\]
and the \(L^p\) Poincar\'e constant at infinity by
\[
\mathcal{E}_p(\Omega):=\sup_{R>0}\lambda_{1,p}(\Omega\setminus \overline{B_R}).
\]
Assume that there exists \(k\in\mathbb{N}\setminus\{0\}\) such that
\[
\lambda_{k,p}^{\rm LS}(\Omega)<\mathcal{E}_p(\Omega).
\]
Here \(\lambda\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\) if there exists \(u\in W^{1,p}_0(\Omega)\setminus\{0\}\) such that \(-\Delta_p u=\lambda |u|^{p-2}u\) weakly in \(\Omega\). Under these assumptions, which statement about the minmax values \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) and their eigenfunctions holds?

For every \(\ell\in\{1,\dots,k\}\), the minmax value \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\). Moreover:

- If \(\mathcal{E}_p(\Omega)<+\infty\), then for every \(\ell\in\{1,\dots,k\}\) there exist constants \(C_\ell=C_\ell(N,p,\Omega)>0\) and \(\alpha_\ell=\alpha_\ell(p,\Omega)>0\) such that every eigenfunction \(u_\ell\in W^{1,p}_0(\Omega)\) associated with \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) satisfies
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha_\ell |x|}\qquad\text{for every }x\in\Omega.
\]
- If \(\mathcal{E}_p(\Omega)=+\infty\), then for every \(\ell\in\mathbb{N}\setminus\{0\}\) and every \(\alpha>0\) there exists a constant \(C_\ell=C_\ell(N,p,\Omega,\alpha)>0\) such that
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha |x|}\qquad\text{for every }x\in\Omega.
\]

Choices:
B.
For every \(\ell\in\{1,\dots,k\}\), the minmax value \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\). Moreover:

- If \(\mathcal{E}_p(\Omega)<+\infty\), then for every \(\ell\in\{1,\dots,k\}\) and every \(\alpha>0\) there exists a constant \(C_\ell=C_\ell(N,p,\Omega,\alpha)>0\) such that every eigenfunction \(u_\ell\in W^{1,p}_0(\Omega)\) associated with \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) satisfies
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha |x|}\qquad\text{for every }x\in\Omega.
\]
- If \(\mathcal{E}_p(\Omega)=+\infty\), then for every \(\ell\in\mathbb{N}\setminus\{0\}\) there exist constants \(C_\ell=C_\ell(N,p,\Omega)>0\) and \(\alpha_\ell=\alpha_\ell(p,\Omega)>0\) such that
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha_\ell |x|}\qquad\text{for every }x\in\Omega.
\]
C. For every \(\ell\in\{1,\dots,k\}\), the minmax value \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\).
D.
For every \(\ell\in\mathbb{N}\setminus\{0\}\), the minmax value \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\). Moreover:

- If \(\mathcal{E}_p(\Omega)<+\infty\), then for every \(\ell\in\mathbb{N}\setminus\{0\}\) there exist constants \(C_\ell=C_\ell(N,p,\Omega)>0\) and \(\alpha_\ell=\alpha_\ell(p,\Omega)>0\) such that every eigenfunction \(u_\ell\in W^{1,p}_0(\Omega)\) associated with \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) satisfies
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha_\ell |x|}\qquad\text{for every }x\in\Omega.
\]
- If \(\mathcal{E}_p(\Omega)=+\infty\), then for every \(\ell\in\mathbb{N}\setminus\{0\}\) and every \(\alpha>0\) there exists a constant \(C_\ell=C_\ell(N,p,\Omega,\alpha)>0\) such that
\[
|u_\ell(x)|\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha |x|}\qquad\text{for every }x\in\Omega.
\]
E.
For every \(\ell\in\{1,\dots,k\}\), the minmax value \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) is an eigenvalue of the Dirichlet \(p\)-Laplacian on \(\Omega\). Moreover:

- If \(\mathcal{E}_p(\Omega)<+\infty\), then for every \(\ell\in\{1,\dots,k\}\) there exist constants \(C_\ell=C_\ell(N,p,\Omega)>0\) and \(\alpha_\ell=\alpha_\ell(p,\Omega)>0\) such that every eigenfunction \(u_\ell\in W^{1,p}_0(\Omega)\) associated with \(\lambda_{\ell,p}^{\rm LS}(\Omega)\) satisfies
\[
\|u_\ell\|_{L^p(\Omega\setminus B_R)}\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha_\ell R}\qquad\text{for every }R>0.
\]
- If \(\mathcal{E}_p(\Omega)=+\infty\), then for every \(\ell\in\mathbb{N}\setminus\{0\}\) and every \(\alpha>0\) there exists a constant \(C_\ell=C_\ell(N,p,\Omega,\alpha)>0\) such that
\[
\|u_\ell\|_{L^p(\Omega\setminus B_R)}\le C_\ell\,\|u_\ell\|_{L^p(\Omega)}\,e^{-\alpha R}\qquad\text{for every }R>0.
\]

dropped the decay conclusions, keeping only eigenvalue existence for indices up to k

range of indices controlled by the assumption below the threshold

pointwise decay conclusion replaced by mere tail \(L^p\)-decay estimate

The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the valid conclusion, without obvious lexical clues pointing uniquely to A.

The item is very close to a theorem-recall question: the stem states the assumptions and asks which conclusion holds. The options do vary meaningful quantifiers and ranges, so it is not a pure verbatim restatement, but it remains only a mild reformulation.

Some reasoning is needed to track the index range and the decay quantifiers in the cases Ep(Omega)<infinity versus Ep(Omega)=infinity. However, the task is mostly recognition of the exact theorem statement rather than genuine derivation, and the presence of a weaker true option reduces real generative pressure.

Several distractors are mathematically plausible and target common errors: swapping quantifiers, overextending from l<=k to all l, and weakening pointwise decay to tail L^p decay. But choice C is a weaker statement that also appears true under the stated assumptions, so the single-correct structure is compromised.

A mathematically sophisticated but only moderate-quality MCQ: it avoids direct leakage and has mostly strong distractor design, yet it is largely theorem-recall and is weakened by an apparently true weaker option, making the item ambiguous.