\label{thvecchio}\cite{CPP}
    Let $\Sigma_D$ be a cone such that $\bar D\subset \mathbb{S}^{N-1}_+$ and $\lambda_1(D)<N-1$, where $\mathbb{S}^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of \eqref{printro} which are also fast decaying, i.e.
    $$
    v(x)=O(|x|^{2-N}) \quad \text{as }|x|\to\infty.
    $$

\begin{corollary}\label{cor_L2}
Let $\alpha\in(0,2)$ be such that $D_{\alpha}\subset\S_+^{N-1}$, and $  \lambda_1(D_{\alpha})<N-1$. Then there exists $\bar\alpha\in(0,\alpha)$ such that
$$
\lambda_1(D_{\bar\alpha})=N-1.
$$
Possibly redefining the parameter $\alpha$, we may assume $\bar\alpha=1$.
\end{corollary}

In order to understand the behavior of $\lambda_1(D_{\alpha})$ as $\alpha$ varies, we interpret the eigenvalue $\lambda_1(D_{\alpha})$ as the first Neumann eigenvalue of a suitable operator on the fixed domain $D$. In fact, $\lambda_1(D_{\alpha})$ is the first eigenvalue of the Laplacian on $D$ with respect to the metric $g_{\alpha}$, the pull-back of the round metric on $D_{\alpha}$ through $\phi_{\alpha}$.

More precisely, if $g$ denotes the standard round metric on $\mathbb S^{N-1}$, consider on $D$ the metric $g_{\alpha}:=\phi_{\alpha}^*g$:
$$
\langle X,Y\rangle_{g_{\alpha}}=\langle d\phi_{\alpha}(X),d\phi_{\alpha}(Y)\rangle_g
$$
for all tangent vectors $X,Y$. Here by $\langle\cdot,\cdot\rangle_h$ we denote the scalar product induced on tangent spaces by a metric $h$. In what follows we will highlight the dependence on the metric only when necessary, and we will often omit it when the metric is the  standard Euclidean metric in $\mathbb R^N$ or the standard round metric on $\mathbb S^{N-1}$.
Then (by construction), the manifold $(D,g_{\alpha})$ is isometric to the manifold $(D_{\alpha},g)$, i.e., the domain $D_{\alpha}$ with the round metric $g$. Therefore
$$
\lambda_1(D_{\alpha})=\lambda_1(D,g_{\alpha})
$$
where the notation $\lambda_1(D,h)$ simply stands for the first Neumann eigenvalue of the Laplacian $\Delta_h$ on $D$ for a metric $h$.
The Laplacian commutes with isometries, that is
$$
\Delta_{g_{\alpha}}(\phi_{\alpha}^*f)=\phi_{\alpha}^*(\Delta_gf)(=\phi_{\alpha}^*(\Delta_{\mathbb S^{N-1}}f))
$$
and this is a concrete way of looking at the new operator on the fixed domain $D$.

If $D\subset\mathbb S^{N-1}_+$, then $r<\pi/2$ for any point of $D$ ($r$ is the polar coordinate, i.e., the distance from the north pole). The metric $g_{\alpha}$ on $D$, for $\alpha\in(0,2)$, is almost isometric to an homothety of the spherical metric on $D$, in fact
$$
\min\{1,\frac{\sin(\alpha\frac{\pi
}{2} )}{\alpha}\}<\frac{\sin(\alpha r)}{\alpha\sin(r)}<\max\{1,\frac{\sin(\alpha\frac{\pi
}{2} )}{\alpha}\}
$$
and then we write
\beq\label{quasiisom}
g_{\alpha}=\alpha^2dr^2+\sin^2(\alpha r)g_{\mathbb S^{N-2}}\approx\alpha^2(dr^2+\sin^2( r)g_{\mathbb S^{N-2}})=\alpha^2 g.
\eeq

\subsection{The induced diffeomorphisms on the cone}

We pass now to the cone  $\Sigma_{D_{\alpha}}$, which we want to recast to the fixed cone $\Sigma_{D}$. Here, we are considering spherical coordinates in $\mathbb R^N$; in particular, we denote by $\rho$ the distance from the origin in $\mathbb R^N$, that is, the radial coordinate. A point $x\in\Sigma_D$ will have then coordinates $(\rho,p)\in(0,\infty)\times D_{\alpha}$. The standard round metric on $\mathbb S^{N-1}$ is denoted by $g$. The Euclidean metric $G$ on $\mathbb R^N$ (i.e., the metric on $\Sigma_{D_{\alpha}}$) in the coordinates $(\rho,p)$ is
$$
G=d\rho^2+\rho^2 g
$$
Now we consider the map $\Phi_{\alpha}:\Sigma_{D}\to \Sigma_{D_{\alpha}}$  defined by
\beq\label{Phidef}
\Phi_{\alpha}(\rho,p)=(\rho,\phi_{\alpha}(p))
\eeq
and again we consider on $\Sigma_{D}$ the metric $G_{\alpha}$, i.e., the pull-back of the Euclidean metric $G$ though $\Phi_{\alpha}$. Precisely, we have
$$
G_{\alpha}:=\Phi_{\alpha}^*G=d\rho^2+\rho^2\phi_{\alpha}^*g=d\rho^2+\rho^2g_{\alpha}.
$$
Now we express the Laplacian on $\Sigma_{D}$ with respect to the metric $G_{\alpha}$:
\begin{equation}\label{laplacian_new_metric}
\Delta_{G_{\alpha}}=\partial^2_{\rho\rho}u+\frac{N-1}{\rho}\partial_\rho u+\frac{1}{\rho^2}\Delta_{g_{\alpha}}u.
\end{equation}
Again, we have that
\begin{equation}\label{commuting}
\Delta_{G_{\alpha}}(\Phi_{\alpha}^*f)=\Phi_{\alpha}^*(\Delta_Gf)(=\Phi_{\alpha}^*(\Delta f)),
\end{equation}
and $\Delta_G=\Delta$ is the usual Laplacian with respect to the Euclidean metric $G$.

\subsection{Derivative of $\lambda_1(D_{\alpha})$ with respect to $\alpha$}

Now we compute and estimate the derivative
$\frac{d}{d\alpha}\lambda_1(D_{\alpha})\big|_{\alpha=1}=\frac{d}{d\alpha}\lambda_1(D,g_\alpha)\big|_{\alpha=1}$ at a simple eigenvalue. Recall that if $\lambda_1(D_1)$ is simple, then $\lambda_1(D_{\alpha})$ is simple for $\alpha\in(1-\epsilon,1+\epsilon)$ for small $\epsilon$.

\begin{proposition}\label{3.3}
Assume that $\lambda_1(D)=\lambda_1(D_1)$ is simple, and let $u$ be an eigenfunction of $\lambda_1(D)$ normalized by $\int_{D}u^2=1$. Then
\begin{multline}\label{derivative}
\frac{d}{d\alpha}\lambda_1(D_{\alpha})|_{\alpha=1}
=\frac{d}{d\alpha}\lambda_1(D,g_{\alpha})|_{\alpha=1}\\
=\int_{D}\left[(-1+(N-2)\frac{r}{\tan(r)})(\partial_ru)^2+(1+(N-4)\frac{r}{\tan(r)})\frac{|\nabla_{\mathbb S^{N-2}}u|^2}{\sin^2(r)}\right]\sin^{N-2}(r)drd\sigma_{\mathbb S^{N-2}}\\
-\lambda_1(D)\int_{D}u^2(1+(N-2)\frac{r}{\tan(r)})\sin^{N-2}(r)drd\sigma_{\mathbb S^{N-2}}.
\end{multline}
\end{proposition}
To clarify the notation of \eqref{derivative}, here we are considering polar coordinates $(r,\omega)\in(0,\pi/2)\times\mathbb S^{N-2}$ in $\mathbb S^{N-1}_+$ centered at the north pole, and $d\sigma_{\mathbb S^{N-2}}=d\sigma_{\mathbb S^{N-2}}(\omega)$ is the volume element of $\mathbb S^{N-2}$.

The proof of the proposition is standard and consists of deriving the eigenvalue of the Laplacian with respect to a $1$-parameter family of varying metrics, in our case, $g_{\alpha}$, on the fixed manifold $D$. We give the proof following the classical one, which can be found in Berger \cite{berger}.

\begin{proof}[Proof of Proposition \ref{3.3}]
Following \cite[Section 3]{berger}, we find that
$$
\frac{d}{d\alpha}\lambda_1(D,g_{\alpha})|_{\alpha=1}=-\int_{D}\langle q(u),h\rangle_g d\sigma_g,
$$
where $h=\frac{d}{d\alpha}{g_{\alpha}}_{|_{\alpha=1}}$ and $q$ is the quadratic form defined as
$$
q(u)=du\otimes du-\frac{1}{2}(|\nabla u|^2-\lambda_1(D)u^2)g.
$$
The notation $\langle\cdot,\cdot\rangle_g$ stands for the inner product on tensors induced by the metric $g$ and $\otimes$ is the tensor product. We recall that here $g=g_1$ is the standard round metric.

Now we have that
$$
g=dr^2+\sin^2(r)g_{\mathbb S^{N-2}},
$$
$$
h=2dr^2+2r\sin(r)\cos(r)g_{\mathbb S^{N-2}},
$$
and
$$
du\otimes du=(\partial_ru)^2dr\otimes dr+2\frac{\partial_ru}{\sin(r)}dr\otimes d_{\mathbb S^{N-2}}u+\frac{d_{\mathbb S^{N-2}}u\otimes d_{\mathbb S^{N-2}}u}{\sin^2(r)}.
$$
All these objects are $(0,2)$ tensors, and then we compute
$$
\langle g,h\rangle_g=2+2(N-2)\frac{r}{\tan(r)}
$$
and
$$
\langle du\otimes du,h\rangle_g=2(\partial_ru)^2+2\frac{r}{\tan(r)}\frac{|\nabla_{\mathbb S^{N-2}}u|^2}{\sin^2(r)}.
$$
Here we have used that $|\nabla_{\mathbb S^{N-2}}u|^2=|d_{\mathbb S^{N-2}}u|^2$. Recalling that $|\nabla u|^2=(\partial_ru)^2+\frac{|\nabla_{\mathbb S^{N-2}}u|^2}{\sin^2(r)}$ and that $d\sigma_g=\sin^{N-2}(r)drd\sigma_{\mathbb S^{N-2}}$ we get formula \eqref{derivative}.
\end{proof}

Our aim is to show that the derivative \eqref{derivative} is negative under suitable assumption on $D$.

First, note that $\frac{r}{\tan(r)}\in(0,1)$ when $r\in(0,\pi/2)$ ($\frac{r}{\tan(r)}\to 0$ as $r\to\pi/2$, $\frac{r}{\tan(r)}\to 1$ as $r\to 0$, and it is decreasing in $r$).

\medskip

\begin{proposition}
    Let $\lambda_1(D)=\lambda_1(D_1)$ be simple and assume that $D\subset C_t$ where $C_t$ is a geodesic disk of radius $t\in(0,\pi/2)$ such that $t/\tan (t)\ge\frac{N-4}{N-2}$. Then
\beq\label{derneg}
\frac{d
}{d\alpha}\lambda_1(D_\alpha)|_{\alpha=1}<0.
\eeq
\end{proposition}

\begin{proof}
    From \eqref{derivative} and the fact that $\frac{r
    }{\tan r}<1$ for $r\in(0,\pi/2)$ we have
\begin{multline}\label{derivative2}
\frac{d}{d\alpha}\lambda_1(D_{\alpha})|_{\alpha=1}
\\
<\int_{D}\left[(N-3)\left((\partial_ru)^2+\frac{|\nabla_{\mathbb S^{N-2}}u|^2}{\sin^2(r)}\right)-\lambda_1(D)u^2((N-2)b+1)\right]\sin^{N-2}(r)drd\sigma_{\mathbb S^{N-2}}\\
=(N-3)\int_{D}|\nabla u|^2-\left(\frac{(N-2)b+1}{N-3}\right)\lambda_1(D)u^2\sin^{N-2}(r)drd\sigma_{\mathbb S^{N-2}},
\end{multline}
    where $b=b(r)=r/\tan r$.
Then taking $t$ such that $t/\tan t\ge\frac{N-4}{N-2}$ by the monotonicity of the function $t/\tan t$ we have that
\beq\label{contob}
b=\frac{r}{\tan r}\ge \frac{t}{\tan t}\ge \frac{N-4}{N-2}\quad \forall r\in (0,t),
\eeq
and hence
$$
\frac{(N-2)b+1}{N-3}>1.
$$
Since
$$
\lambda_1(D)=\frac{\int_D |\nabla_{\mathbb S^{N-1}} u|^2}{\int_D u^2},
$$
we obtain \eqref{derneg}.

\end{proof}
\begin{remark}
Note that for $N=3,4$ one can see directly from \eqref{derivative2} or from \eqref{contob} that the derivative is negative for any  domain $D_1=D\subset\mathbb S^{N-1}_+$. Instead for $N\geq 5$ we have to consider domains contained in smaller disks.
\end{remark}

\subsection{Proof of Proposition \ref{limit_L2}}\label{ss3.3}
We prove here Proposition \ref{limit_L2}. In fact, one can see that $g_{\alpha}$ is quasi-isometric to a homothety of the round metric $g$ of factor $\alpha^2$, and the eigenvalues can be compared.

\begin{proof}[Proof of Proposition \ref{limit_L2}]

Observe that from \eqref{quasiisom} on $D$ the metric $g_{\alpha}$ and the metric $\alpha^2 g$ are quasi-isometric. This means that there exists a constant $K>0$ such that
\beq\label{compar}
\frac{1}{K}<\frac{g_{\alpha}}{\alpha^2 g}< K
\eeq
For any eigenvalue $\lambda_j$, by the min-max principle we have
$$
\lambda_j(D,g_\alpha) = \min_{\substack{V\subset H^1(D)\\\dim V = j}} \max_{u \in V, u \neq 0} \frac{\int_{D} |\nabla_{g_{\alpha}} u|_{g_{\alpha}}^2 \, d\sigma_{g_{\alpha}}}{\int_{D} u^2 \, d\sigma_{g_{\alpha}}}.
$$
Here $d\sigma_{g_{\alpha}}$ is the Riemannian volume element of the metric $g_{\alpha}$ and $|\nabla_{g_{\alpha}}u|_{g_{\alpha}}$ is the norm of the gradient of $u$ for the metric $g_{\alpha}$.
From \eqref{compar} we deduce
$$
d\sigma_{\alpha^2g} = \sqrt{\det \alpha^2g} \, d\theta_1 \dots d\theta_{N-1} > \frac{1}{K^{\frac{N-1}{2}}} \sqrt{\det g_\alpha} \, d\theta_1 \dots d\theta_{N-1} = \frac{1}{K^{\frac{N-1}{2}}} d\sigma_{g_\alpha}
$$
and
$$
d\sigma_{\alpha^2g}<K^{\frac{N-1}{2}} d\sigma_{g_\alpha}.
$$
Moreover,
$$
\frac1K|\nabla_{g_\alpha} u|^2_{g_\alpha}<|\nabla_{\alpha^2g} u|^2_{\alpha^2g}<K|\nabla_{g_\alpha} u|^2_{g_\alpha}.
$$
Then

$$\frac{\int_{D} |\nabla_{g_{\alpha}} u|_{g_{\alpha}}^2 \, d\sigma_{g_{\alpha}}}{\int_{D} u^2 \, d\sigma_{g_{\alpha}}}<K^{1+(N-1)/2+(N-1)/2}\frac{\int_{D} |\nabla_{\alpha^2g} u|_{\alpha^2g}^2 \, d\sigma_{\alpha^2g}}{\int_{D} u^2 \, d\sigma_{\alpha^2g}}
$$

and
$$\frac{1}{K^{1+(N-1)/2+(N-1)/2}}\frac{\int_{D} |\nabla_{\alpha^2g} u|_{\alpha^2g}^2 \, d\sigma_{\alpha^2g}}{\int_{D} u^2 \, d\sigma_{\alpha^2g}}<\frac{\int_{D} |\nabla_{g_{\alpha}} u|_{g_{\alpha}}^2 \, d\sigma_{g_{\alpha}}}{\int_{D} u^2 \, d\sigma_{g_{\alpha}}}.
$$
Then, from the Min-Max Theorem, we conclude that
$$
\frac{1}{K^N}\lambda_j(D,\alpha^2g)<\lambda_j(D,g_\alpha)<K^N\lambda_j(D,\alpha^2g).
$$

Now, we have that
$$
\lambda_j(D,\alpha^2 g)=\frac{1}{\alpha^2}\lambda_j(D,g)=\frac{1}{\alpha^2}\lambda_j(D)
$$
and this yields the result.

\end{proof}

\subsection{An example of simple eigenvalue $N-1$}

We provide here an example of a domain $D$ contained in an hemisphere in $\R^3$ for which the first eigenvalue $\lambda_1(D)$ is $2$ and is simple. We just start from a dumbbell domain on $\mathbb S^2_+$. Assume it is symmetric with respect to the equator and to the (arc of) great circle $\omega=0$. Here we are using polar coordinates $( r ,\omega)$, with $ r $ the distance from the north pole. See Figure \ref{fig1}. Roughly speaking, $D$ consists of two disjoint `fat' parts (e.g., two geodesic disks) connected by a thin channel. In our example, the fat parts are symmetric with respect to the equator (see Figure \ref{fig1}). The domain $D$ is contained in the strip $ r \in( r _0,\pi- r _0)$, $ r _0\in(0,\pi/2)$. For any $ r '\in( r _0,\pi- r _0)$, we call $\gamma_{ r '}:=D\cap\{ r = r '\}$ (see Figure \ref{fig1}). Assume that $\max_{ r \in( r _0,\pi- r _0)}|\gamma_{ r }|<\pi/2$.

From Corollary \ref{cor_L2} and from the fact that the first eigenvalue of a fixed dumbbell can be arbitrarily small when the width of the channel is small (see e.g., \cite{IPW}), we deduce that we can realize such a domain $D=D_1$ with $\lambda_1(D_1)=2$. We show that $\lambda_1(D_1)=2$ it is simple. It will follow that $\lambda_1(D_{\alpha})$  remains simple for all $\alpha\in(1-\varepsilon,1+\varepsilon)$ for some $\varepsilon>0$.

 Since we are dealing with $\lambda_1(D)$, its multiplicity is at most $2$ by Courant's Theorem.  Since any corresponding eigenfunction has exactly two nodal domains, by symmetry  we know that the eigenspace is spanned by eigenfunctions that are either odd with respect to the equator and even with respect to the arc $\omega=0$, or the other way around. Note that an eigenfunction $u$ corresponding to $\lambda_1(D)$ cannot be odd (respectively even) with respect to both the equator and the great circle $\omega=0$. Assume that $u$ is odd with respect to the arc of great circle $\omega=0$ (the yellow arc in Figure \ref{fig1}). Then, for any $ r \in( r _0,\pi- r _0)$, $\int_{\gamma_{ r }}u=0$. Therefore, $\int_{\gamma_{ r }}(\partial_{\omega}u)^2\geq\frac{\pi^2}{|\gamma_{ r }|^2}\int_{\gamma_{ r }}u^2$. Here $\frac{\pi^2}{|\gamma_{ r }|^2}$ is the first (non trivial) Neumann eigenvalue on $\gamma_{ r }$.

By assumption we have $|\gamma_{ r }|<\pi/2$. This implies that $\int_{\gamma_{ r }}(\partial_{\omega}u)^2\geq 4\int_{\gamma_{ r }}u^2$, which means $\int_{\gamma_{ r }}|\nabla u|^2\geq 4\int_{\gamma_{ r }}u^2$. Integrating this last inequality in $ r \in( r _0,\pi- r _0)$ we get that $\int_D|\nabla u|^2\geq 4\int_Du^2$. Hence we have a contradiction since $\lambda_1(D)=2$. Therefore the first eigenvalue is simple and a corresponding eigenfunction is odd with respect to the equator.

The construction is valid in any space dimension, we just needed to consider a rotationally symmetric dumbbell in $\mathbb S^{N-1}$ foliated by $N-2$ dimensional spheres of small enough maximal volume.

\begin{figure}
\includegraphics[width=0.9\textwidth]{fig_foliation.pdf}
\caption{}
\label{fig1}
\end{figure}

\section{Bifurcation results}\label{sec3}
\subsection{Spectral analysis}\label{spectan}
Let $D\subset\S^{N-1}_+$ be a smooth domain and assume that the corresponding cone $\Sigma_D$ is a Lipschitz domain. We consider the map $\phi_\alpha, \alpha \in (0,2)$ defined in Section \ref{ss3.1} and the cone $\Sigma_{D_\alpha}$.
Consider the bubble $U\in\mathcal{
D
}^{1,2}(\Sigma_{D_\alpha})$:
\beq\label{bubbledef}
U(x):=c_0(1+|x|^2)^{-\frac{N-2}{2}}\qquad x\in \Sigma_{D_\alpha},
\eeq
where $c_0$ is a constant such that
$$
-U^{2-2^*}\Delta U=U\qquad \text{in} \quad \Sigma_{D_\alpha}.
$$
The rescaled bubbles will be denoted by $$
U_{s}(x):=s U(s^{\frac{2}{N-2}}x).
$$
\begin{lemma}\label{lemma_spettro}
The operator $L_{U}=U^{2-2^*}\Delta$ on $L^2(U^{2^*-2})$ has discrete spectrum.
\end{lemma}
\begin{proof}
    The proof can be found in \cite{CPP2}.
\end{proof}
To describe the spectrum of the operator $L_{U}$ we need to know the eigenvalues $\lambda_j(D_\alpha), j\in \N$, of problem \eqref{pb_lambda_def1} on $D_\alpha$.
In particular, we recall that the  $\lambda_0(D_\alpha)=0$ and the corresponding eigenfunction is constant, while the first positive eigenvalue is $\lambda_1(D_\alpha)$.

Now we are ready to address the following eigenvalue problem that is studied in \cite{CPP2},
\begin{equation}\label{pb_mu_def2}
\begin{cases}
\Delta v+\mu_{\alpha} U^{2^*-2}v=0 &\text{in } {\Sigma_{D_\alpha}}\\
\partial_\nu v=0 &\text{on }  \partial{\Sigma_{D_\alpha}}\setminus\{0\} \\
v\in \mathcal{D}^{1,2}(\Sigma_{D_\alpha}), &
\end{cases}
\end{equation}
whose  eigenvalue will be denoted by $\mu_{i,\alpha}, i\in\N_+$. The eigenvalue equation in \eqref{pb_mu_def2} can be written by using spherical coordinates, and we can look for solutions by using separation of variables: let $(\rho,\theta)\in(0,+\infty)\times\mathbb S^{N-1}$ the standard spherical coordinates in $\mathbb R^N$ and let $v=R(\rho)Y(\theta)$ be a solution of \eqref{pb_mu_def2} for an eigenvalue $\mu_\alpha$, then it holds that $Y$ is an eigenfunction of \eqref{pb_lambda_def1} on $D_\alpha$ for some $j$, while $R$ satisfies (see \cite{CPP2}, Proposition 2.6)
\begin{equation} \label{SL_eq}
\begin{cases}
(\rho^{N-1}R')'+\rho^{N-1}(-\lambda_j(D_\alpha) \rho^{-2}+\mu_\alpha k_0^{2^*-2}(1+\rho^2)^{-2})R=0 & \text{in }\R_+ \\
\lim_{\rho\to 0}\rho^{N-1}R'(\rho)=0 & \\
R(\rho)=o(\rho^{2-N+\epsilon})  \quad \forall\epsilon> 0\text{ as }\rho\to + \infty \,. &

\end{cases}
\end{equation}
We denote by $\mu_{k,\lambda_j(D_\alpha)}, k\in\N_+, j\in\N$, the sequence of eigenvalues of \eqref{SL_eq} obtained in correspondence of $\lambda_j(D_\alpha)$ and denote by $R^j_k$ the corresponding eigenfunctions.

For $\lambda_0(D_\alpha)=0$ the eigenvalues are given by
\beq\label{lambdazerok}\mu_{k,\lambda_0}=\bigg((k-1)(k+N-2)\frac{4}{N(N-2)}+1\bigg)\,.
\eeq
The eigenvalues associated with $\lambda_1(D_\alpha)$ are
\begin{equation}\label{betaeigenvalue}
\mu_{1,\lambda_1}= \frac{1}{N(N-2)}\sqrt{(N-2)^2 + 4\lambda_1(D_\alpha)}( 2 +\sqrt{(N-2)^2 + 4\lambda_1(D_\alpha)}) .
\end{equation}
In details, it holds the following result (see \cite{CPP2}).

\begin{proposition} \label{lemma_autov}

Let $D_\alpha\subset\S_+^{N-1}$ and $m\in\N_+$ be such that
\beq\label{lemma_autov_cond}
(m-1)(m+N-3)<\lambda_1(D_\alpha)\le m(m+N-2).
\eeq
Then each eigenvalue $\mu_{i,\alpha}$ of \eqref{pb_mu_def2}, with $i=1,...,m$,
is simple and is given by
$$\mu_{i,\alpha}=\bigg((i-1)(i+N-2)\frac{4}{N(N-2)}+1\bigg),
$$
i.e. are the eigenvalues corresponding to $\lambda_0$ in \eqref{SL_eq} while
$$
\mu_{m+1,\alpha}=\frac{1}{N(N-2)}\sqrt{(N-2)^2 + 4\lambda_1(D_\alpha)}\bigg( 2 +\sqrt{(N-2)^2 + 4\lambda_1(D_\alpha)}\bigg).
$$
Moreover, $\psi_i=R_i(r)Y_0(\theta)$ is an eigenfunction of \eqref{pb_mu_def2} corresponding to $\mu_{i,\alpha}$, for any $i=1,...,m,$ where
$$
R_i=\frac{P_{i}(|x|^2)}{(1+|x|^2)^{\frac{N-2}{2}}}
$$
and $P_i(|x|^2)$ are given as in \cite[Appendix]{CPP2}. Instead an eigenfunction corresponding to $\mu_{m+1,\alpha}$ is given by  $\psi_{m+1}=R_{m+1}Y_1(\theta)$ where
$$R_{m+1}=(|x|^2+1)^{1-\frac{N}{2}-\beta(\lambda_1(D_\alpha))}|x|^{\beta(\lambda_1(D_\alpha))}$$
with
$2\beta(\lambda_1(D_\alpha))=-(N-2) + \sqrt{(N-2)^2 + 4\lambda_1(D_\alpha)} $ and $Y_1$ is an eigenfunction of \eqref{pb_lambda_def1} corresponding to $\lambda_1(D_\alpha)$.
\end{proposition}
\begin{proof}
    The proof follows by \cite[Proposition 2.7]{CPP2} and the fact that we are taking $U$ such that $-\Delta U=U^{2^*-1}$, so that the constant $S^{2^*}_U$ in \cite{CPP2} is replaced by $1$.
\end{proof}

\begin{corollary}\label{cor_autoval}
    Let $D_{\bar{\alpha}}\subset\S_+^{N-1}$ be such that $\lambda_1(D_{\bar\alpha})=N-1$, then
    $$
    \mu_{1,{\bar\alpha}}=1
    $$
    and the first eigenfunction is given by $U_{\bar\alpha}
    $,
$$\mu_{2,{\bar\alpha}}= \frac{1}{N(N-2)}\sqrt{(N-2)^2 + 4\lambda_1(D_{\bar\alpha})}\bigg( 2 +\sqrt{(N-2)^2 + 4\lambda_1(D_{\bar\alpha})}\bigg)=
2^*-1
$$
and the second eigenfunctions are given by $\partial_s U_{s,{\bar\alpha}}$ and $\psi_2=(|x|^2+1)^{-\frac{N}{2}}|x|Y_1(\frac{x}{|x|})$.
\end{corollary}

\begin{theorem}[Crandall-Rabinowitz Bifurcation Theorem] \label{CRT}
Let $X$ and $Y$ be Banach spaces, and let $\Omega\subset X$ and $I \subset \mathbb{R}$ be open domains, where we assume $0 \in \Omega$. Denote the elements of $\Omega$ by $v$ and the elements of $I$ by $t$. Let $F: I \times \Omega \to Y$ be a $C^2$ operator such that

\begin{itemize}
    \item[(i)] $F(t, 0) = 0$ for all $t \in I$,
    \item[(ii)] $\text{Ker} D_v F(t_*, 0) = \mathbb{R}w$ for some $t_* \in I$ and some $w \in X \setminus \{0\}$;
    \item[(iii)] $\text{codim} \, \text{Im} D_v F(t_*, 0) = 1$;
    \item[(iv)] $D_t D_v F(t_*, 0)(w) \notin \text{Im} D_v F(t_*, 0)$.
\end{itemize}

Then there exists a nontrivial $C^1$ curve
$$\quad (-\epsilon, \epsilon) \ni s \mapsto (t(s), v(s)) \in I \times X,
$$
for some $\epsilon > 0$, such that:
\begin{itemize}
    \item[(1)] $t(0) = t_*, \, t'(0) = 0, \, v(0) = 0, \, v'(0) = w$;
    \item[(2)] $F(t(s), v(s)) = 0$ for all $s \in (-\epsilon, +\epsilon)$.
\end{itemize}

Moreover, there exists a neighborhood $N$ of $(t_*, 0)$ in $X \times Y$ such that all solutions of the equation $F(t, v) = 0$ in $N$ belong to the trivial solution line $\{(t, 0)\}$ or to the curve. The intersection $(t_*, 0)$ is called a bifurcation point.

\end{theorem}

\begin{cases}
    -\Delta u=u^{2^*-1}&\text{in }\Sigma_D,\\
    \frac{\partial u}{\partial\nu}=0&\text{on }\partial\Sigma_D\setminus\{0\},\\
    u>0&\text{in }\Sigma_D,
\end{cases}
\eeq
where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\mathcal{D}^{1,2}(\Sigma_D)$ into $L^{2^*}(\Sigma_D)$ and
$$
\mathcal{D}^{1,2}(\Sigma_D)=\{u\in L^{2^*}(\Sigma_D):|\nabla u|\in L^2(\Sigma_D)\}.
$$
It is well known that the radial function:
\beq\label{bubbleintro}
U(x)=c_0\bigg(\frac{1}{1+|x|^2}\bigg)^\frac{N-2}{N},\qquad x\in\Sigma_D,
\eeq
is a solution of \eqref{printro}, for $c_0=(N(N-2))^\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\R^N$, the function $U$, its rescalings and translations are the only positive solutions of \eqref{printro}. Obviously they are also the only radial solutions of \eqref{printro} and we refer to them as the trivial solutions of \eqref{printro}. Then the question is whether in $\Sigma_D$, for $D\ne\S^{N-1}$, there exist also nonradial solutions.

The study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \cite{LP,LPT}).
In particular in \cite{LPT} it was proved that if the cone $\Sigma_D$ is convex then the bubbles are the only positive solution of \eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \cite{CFR}. Thus the question is whether nonradial solutions of \eqref{printro} exist in nonconvex cones.

Since minimizers of the Sobolev quotient
\beq\label{Sobqintro}
Q_D(u)=\frac{\big(\int_{\Sigma_D}|\nabla u|^2\big)^\frac{1}{2}}{\big(\int_{\Sigma_D} |u|^{2^*}\big)^\frac{1}{2^*}},\qquad u\in\mathcal{D}^{1,2}(\Sigma_D),u\ne 0,
\eeq
produce solutions of \eqref{printro}, a first result was obtained in \cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \eqref{printro} admits a nonradial solution (as a minimizer of \eqref{Sobqintro}) was obtained in \cite{CPP}.
It pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\Delta_{\S^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.
The result of \cite{CPP} is the following:
\begin{theorem}\label{thvecchio}\cite{CPP}
    Let $\Sigma_D$ be a cone such that $\bar D\subset \S^{N-1}_+$ and $\lambda_1(D)<N-1$, where $\S^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of \eqref{printro} which are also fast decaying, i.e.
    $$
    v(x)=O(|x|^{2-N}) \quad \text{as }|x|\to\infty.
    $$
\end{theorem}
It is interesting to quote that the same bound on the eigenvalue $\lambda_1(D)$, i.e. $\lambda_1(D)<N-1$, is also the key point to show breaking of symmetry results for overdetermined and constant mean curvature problems in cones (see \cite{IPW}).

The proof of Theorem \ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \eqref{Sobqintro} as soon as $\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \cite{CPP}). Instead when $\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \cite{CPP} shows that, varying the domains $D\subset \S^{N-1}$ (and hence the cone $\Sigma_D$) the bubble $U$ is degenerate for any cone $\Sigma_{D_1}$ such that $\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.

This is the aim of the present paper. To state precisely our result, let $D=D_1$ be a domain in $\S^{N-1}_+$ such that $\lambda_1(D_1)=N-1$.
We consider the family of domains $D_\alpha$ which are diffeomorphic to $D_1$ through the diffeomorphism $\Phi_\alpha$ defined in Section \ref{sec2} (see \eqref{Phidef}), for $\alpha\in (0,\alpha^*)$, where $\alpha^*>0$ is the maximum value of $\alpha$ such that $\Phi_\alpha(D_1)$ is contained in $\S^{N-1}_+$. By construction, we have that $\alpha^*>1$.
We stress that, for every $\alpha$ the standard bubble $U$ is a solution of \eqref{printro} in $\Sigma_{D_\alpha}$, so we have the trivial curve of solutions $\alpha
\to(\alpha,U)$, for every $\alpha\in(0,\alpha^*)$. Our bifurcation result is the following.
\begin{theorem}\label{mainth}
Let $\Sigma_{D_\alpha}$ be a family of cones as above, $\alpha\in(0,\alpha^*)$ and assume that $\lambda_1({D_1})=N-1$ is a simple eigenvalue. Then the point $(1,U)$ is a bifurcation point for the trivial curve $\alpha\mapsto(\alpha,U)$ of solutions to \eqref{printro}. More precisely there exist $\epsilon>0$ and a $C^1$ curve  $(-\epsilon,\epsilon)\ni s\to (\alpha_s,v_s)$,  such that $\alpha_0=1$, $v_0=U$, $D_{\alpha_s}\ne D_{\alpha_1}$ and
$$
\begin{cases}
   -\Delta v_s=v_s^{2^*-1} &\text{in }\Sigma_{D_{\alpha_s}}\\
    v_s\in \mathcal{{D}}^{1,2}(\Sigma_{D_{\alpha_s}}),
\end{cases}

Let $D$ be a smooth domain on the unit sphere $\mathbb{S}^{N-1}\subset\mathbb R^N$, with $N\geqslant 3,$ and let $\Sigma_D\subset\mathbb R^N$ be the cone spanned by $D$:
\begin{equation}\label{cono}
\Sigma_D=\{x\in\mathbb R^N: x=\rho p,p\in D,\rho\in(0,+\infty)\},
\end{equation}
and assume that $\Sigma_D$ is a Lipschitz domain.
We consider the following semilinear critical Neumann problem
\begin{equation}\label{printro}
\begin{cases}
 -\Delta u=u^{2^*-1}&\text{in }\Sigma_D,\\
 \frac{\partial u}{\partial\nu}=0&\text{on }\partial\Sigma_D\setminus\{0\},\\
 u>0&\text{in }\Sigma_D,
\end{cases}
\end{equation}
where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent for the immersion $\mathcal{D}^{1,2}(\Sigma_D)$ into $L^{2^*}(\Sigma_D)$ and
$$
\mathcal{D}^{1,2}(\Sigma_D)=\{u\in L^{2^*}(\Sigma_D):|\nabla u|\in L^2(\Sigma_D)\}.
$$
It is well known that the radial function:
\begin{equation}\label{bubbleintro}
U(x)=c_0\bigg(\frac{1}{1+|x|^2}\bigg)^\frac{N-2}{N},\qquad x\in\Sigma_D,
\end{equation}
is a solution of \eqref{printro}, for $c_0=(N(N-2))^\frac{N-2}{4}$, as well as any of its rescaling. We recall that $U$ is a minimizer for the Sobolev quotient in the whole $\mathbb R^N$ and we refer to it as the standard bubble. Moreover, in the case of $\mathbb R^N$, the function $U$, its rescalings and translations are the only positive solutions of \eqref{printro}. Obviously they are also the only radial solutions of \eqref{printro} and we refer to them as the trivial solutions of \eqref{printro}. Then the question is whether in $\Sigma_D$, for $D\ne\mathbb{S}^{N-1}$, there exist also nonradial solutions.

The study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \cite{LP,LPT}).
In particular in \cite{LPT} it was proved that if the cone $\Sigma_D$ is convex then the bubbles are the only positive solution of \eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \cite{CFR}. Thus the question is whether nonradial solutions of \eqref{printro} exist in nonconvex cones.

Since minimizers of the Sobolev quotient
\begin{equation}\label{Sobqintro}
Q_D(u)=\frac{\big(\int_{\Sigma_D}|\nabla u|^2\big)^\frac{1}{2}}{\big(\int_{\Sigma_D} |u|^{2^*}\big)^\frac{1}{2^*}},\qquad u\in\mathcal{D}^{1,2}(\Sigma_D),u\ne 0,
\end{equation}
produce solutions of \eqref{printro}, a first result was obtained in \cite{CP} by showing that nonradial minimizers exist in some nonconvex cones. Later a more precise characterization of a class of nonconvex cones for which \eqref{printro} admits a nonradial solution (as a minimizer of \eqref{Sobqintro}) was obtained in \cite{CPP}.
It pointed out the crucial role played, in the break of symmetry, by the eigenvalue $\lambda_1(D)$ which is the first nonzero eigenvalue of the Laplace-Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$, with homogeneous Neumann boundary conditions.
The result of \cite{CPP} is the following:

The study of solutions of the critical equation in cones dates back to 1988 where it arised in connection with the Sobolev and Isoperimetric inequality in cones (see \cite{LP,LPT}).
In particular in \cite{LPT} it was proved that if the cone $\Sigma_D$ is convex then the bubbles are the only positive solution of \eqref{printro}. More recently this symmetry result was extended to critical equations for more general operators in \cite{CFR}. Thus the question is whether nonradial solutions of \eqref{printro} exist in nonconvex cones.

The proof of Theorem \ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \eqref{Sobqintro} as soon as $\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \cite{CPP}). Instead when $\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \cite{CPP} shows that, varying the domains $D\subset \S^{N-1}$ (and hence the cone $\Sigma_D$) the bubble $U$ is degenerate for any cone $\Sigma_{D_1}$ such that $\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.

Let $D\subset\mathbb S^{N-1}_+$ be a smooth domain. By $\lambda_j(D)$ with $j\in \N$, we denote the eigenvalues of the Laplacian $\Delta_{\mathbb S^{N-1}}$ on $D$ with Neumann boundary conditions, i.e. we consider the following problem:
\begin{equation}\label{pb_lambda_def1}
\begin{cases}
-\Delta_{\S^{N-1}}Y_j = \lambda_j(D) Y_j & \text{on } D\\
\partial_{\nu_D} Y_j = 0 & \text{on } \partial D \,.
\end{cases}
\end{equation}
It is well known that $-\Delta_{\S^{N-1}}$ has a compact, self-adjoint resolvent in $L^2(D)$ and admits a sequence of eigenvalues
\begin{equation} \label{lambdabe}
0=\lambda_0(D)<\lambda_1(D)\le \cdots \le \lambda_j(D) \le \cdots\nearrow+\infty
\end{equation}
(repeated according to their finite multiplicity) and corresponding eigenfunctions $Y_j(\theta) \in L^2(D)$. Here $\theta=(\theta_1,...,\theta_{N-1})$ is the system of coordinates on $D$ induced by the spherical coordinates in $\R^N$. The eigenfunctions $\{Y_j\}_{j\in\mathbb N}$ can be chosen to form a Hilbert basis for $L^2(D)$, they satisfy
\beq\label{def:lapleigbe}
-\Delta_{\S^{N-1}}Y_j(\theta)=\lambda_j(D) Y_j(\theta),\quad \theta\in D,
\eeq
and
$$
\int_{D} \langle \nabla_{\mathbb S^{N-1}} Y_j(\theta) , \nabla_{\mathbb S^{N-1}} Y_i(\theta)\rangle d\sigma(\theta) = \lambda_j(D)\delta_{ij}
$$
where we denote by $\nabla_{\mathbb S^{N-1}}$ the gradient on $\mathbb S^{N-1}$.

\section{Bifurcation results}\label{sec3}
\subsection{Spectral analysis}\label{spectan}
Let $D\subset\S^{N-1}_+$ be a smooth domain and assume that the corresponding cone $\Sigma_D$ is a Lipschitz domain. We consider the map $\phi_\alpha, \alpha \in (0,2)$ defined in Section \ref{ss3.1} and the cone $\Sigma_{D_\alpha}$.
Consider the bubble $U\in\mathcal{
D
}^{1,2}(\Sigma_{D_\alpha})$:
\beq\label{bubbledef}
U(x):=c_0(1+|x|^2)^{-\frac{N-2}{2}}\qquad x\in \Sigma_{D_\alpha},
\eeq
where $c_0$ is a constant such that
$$
-U^{2-2^*}\Delta U=U\qquad \text{in} \quad \Sigma_{D_\alpha}.
$$
The rescaled bubbles will be denoted by $$
U_{s}(x):=s U(s^{\frac{2}{N-2}}x).
$$
\begin{lemma}\label{lemma_spettro}
The operator $L_{U}=U^{2-2^*}\Delta$ on $L^2(U^{2^*-2})$ has discrete spectrum.
\end{lemma}
\begin{proof}
 The proof can be found in \cite{CPP2}.
\end{proof}
To describe the spectrum of the operator $L_{U}$ we need to know the eigenvalues $\lambda_j(D_\alpha), j\in \N$, of problem \eqref{pb_lambda_def1} on $D_\alpha$.
In particular, we recall that the $\lambda_0(D_\alpha)=0$ and the corresponding eigenfunction is constant, while the first positive eigenvalue is $\lambda_1(D_\alpha)$.

Now we are ready to address the following eigenvalue problem that is studied in \cite{CPP2},
\begin{equation}\label{pb_mu_def2}
\begin{cases}
\Delta v+\mu_{\alpha} U^{2^*-2}v=0 &\text{in } {\Sigma_{D_\alpha}}\\
\partial_\nu v=0 &\text{on } \partial{\Sigma_{D_\alpha}}\setminus\{0\} \\
v\in \mathcal{D}^{1,2}(\Sigma_{D_\alpha}), &
\end{cases}
\end{equation}
whose eigenvalue will be denoted by $\mu_{i,\alpha}, i\in\N_+$. The eigenvalue equation in \eqref{pb_mu_def2} can be written by using spherical coordinates, and we can look for solutions by using separation of variables: let $(\rho,\theta)\in(0,+\infty)\times\mathbb S^{N-1}$ the standard spherical coordinates in $\mathbb R^N$ and let $v=R(\rho)Y(\theta)$ be a solution of \eqref{pb_mu_def2} for an eigenvalue $\mu_\alpha$, then it holds that $Y$ is an eigenfunction of \eqref{pb_lambda_def1} on $D_\alpha$ for some $j$, while $R$ satisfies (see \cite{CPP2}, Proposition 2.6)
\begin{equation} \label{SL_eq}
\begin{cases}
(\rho^{N-1}R')'+\rho^{N-1}(-\lambda_j(D_\alpha) \rho^{-2}+\mu_\alpha k_0^{2^*-2}(1+\rho^2)^{-2})R=0 & \text{in }\R_+ \\
\lim_{\rho\to 0}\rho^{N-1}R'(\rho)=0 & \\
R(\rho)=o(\rho^{2-N+\epsilon}) \quad \forall\epsilon> 0\text{ as }\rho\to + \infty \,. &

\subsection{Main results}
Let $D_\alpha$ be a domain as in Section \ref{sec2}.
We consider the following problem
\begin{equation}\label{problem}
\begin{cases}
 -\Delta v=v^{2^*-1} &\text{ in }\Sigma_{D_\alpha}\\
 v\in\mathcal{D}^{1,2}(\Sigma_{D_\alpha}).
\end{cases}
\end{equation}
For the reader convenience we recall the notation for the diffeomorphism considered in Section \ref{sec2}.
 Let $\alpha\in(0,\alpha^*)$, where $\alpha^*$ is the maximum value of $\alpha$ such that $\Phi_\alpha(D_1)\subset \S^{N-1}_+$, and consider the map
$$
\phi_{\alpha}:\mathbb S^{N-1}_+\to C_{\alpha}
$$
defined by
$$
\phi_{\alpha}(r,\omega)=(\alpha r,\omega),
$$
where $C_{\alpha}$ is a (spherical) disk centered at the north pole and radius $\alpha\pi/2$. Then we define
$$
\Phi_{\alpha}(\rho,p)=(\rho,\phi_{\alpha}(p)),
$$
where $(\rho,p)\in(0,+\infty)\times D$, and $p=(r,\omega)\in D$ is a point in $D$. The map $\phi_{\alpha}$ induces
$$
\Phi_{\alpha}^*:\mathcal{D}^{1,2}(\Sigma _{D_\alpha})\to\mathcal{D}^{1,2}(\Sigma _{D_1})
$$
defined by $\Phi_{\alpha}^*u=u\circ \Phi_{\alpha}$. We define an operator $\mathcal L_{\alpha}$ acting on $\mathcal D^{1,2}(\Sigma_{D_1})$ by
$$
\mathcal L_{\alpha}(\Phi_{\alpha}^*v)=\Phi_{\alpha}^*(-\Delta v),
$$
for $v\in\mathcal D^{1,2}(\Sigma_{D_{\alpha}})$ (as usual, the operator is defined in the weak sense). This operator can be described more explicitly: it is $-\Delta_{G_{\alpha}}$ defined in \eqref{laplacian_new_metric}, however we won't need this explicit expression in the following.

It is interesting to quote that the same bound on the eigenvalue $\lambda_1(D)$, i.e. $\lambda_1(D)<N-1$, is also the key point to show breaking of symmetry results for overdetermined and constant mean curvature problems in cones (see \cite{IPW}).

The proof of Theorem \ref{thvecchio} relies on a careful analysis of the Morse index of the standard bubble $U$, which shows that it becomes an unstable critical point of \eqref{Sobqintro} as soon as $\lambda_1(D)$ crosses the value $N-1$ (see Theorem 4.3 in \cite{CPP}). Instead when $\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum. Hence the analysis performed in \cite{CPP} shows that, varying the domains $D\subset \mathbb{S}^{N-1}$ (and hence the cone $\Sigma_D$) the bubble $U$ is degenerate for any cone $\Sigma_{D_1}$ such that $\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\lambda_1(D_1)=N-1.$ In other words a branch of nonradial solutions of \eqref{printro} in domains $D$ close to $D_1$ could emanate from the bubble $U$ in $D_1$.

This is the aim of the present paper. To state precisely our result, let $D=D_1$ be a domain in $\mathbb{S}^{N-1}_+$ such that $\lambda_1(D_1)=N-1$.
We consider the family of domains $D_\alpha$ which are diffeomorphic to $D_1$ through the diffeomorphism $\Phi_\alpha$ defined in Section \ref{sec2} (see \eqref{Phidef}), for $\alpha\in (0,\alpha^*)$, where $\alpha^*>0$ is the maximum value of $\alpha$ such that $\Phi_\alpha(D_1)$ is contained in $\mathbb{S}^{N-1}_+$. By construction, we have that $\alpha^*>1$.
We stress that, for every $\alpha$ the standard bubble $U$ is a solution of \eqref{printro} in $\Sigma_{D_\alpha}$, so we have the trivial curve of solutions $\alpha
\to(\alpha,U)$, for every $\alpha\in(0,\alpha^*)$. Our bifurcation result is the following.
\begin{theorem}\label{mainth}
Let $\Sigma_{D_\alpha}$ be a family of cones as above, $\alpha\in(0,\alpha^*)$ and assume that $\lambda_1({D_1})=N-1$ is a simple eigenvalue. Then the point $(1,U)$ is a bifurcation point for the trivial curve $\alpha\mapsto(\alpha,U)$ of solutions to \eqref{printro}. More precisely there exist $\epsilon>0$ and a $C^1$ curve  $(-\epsilon,\epsilon)\ni s\to (\alpha_s,v_s)$,  such that $\alpha_0=1$, $v_0=U$, $D_{\alpha_s}\ne D_{\alpha_1}$ and
$$
\begin{cases}
   -\Delta v_s=v_s^{2^*-1} &\text{in }\Sigma_{D_{\alpha_s}}\\
    v_s\in \mathcal{{D}}^{1,2}(\Sigma_{D_{\alpha_s}}),
\end{cases}$$
and $v_s$ is positive and nonradial.
\end{theorem}
To prove our result we will use the classical Crandall-Rabinowitz bifurcation theorem (see Theorem \ref{CRT}). The main difficulty in proving Theorem \ref{mainth} is to find a good family of domains $D_\alpha$ on $\mathbb{S}^{N-1}$ for which the corresponding eigenvalues $\lambda_1(D_\alpha)$ can be analyzed. Roughly speaking, the goal is to produce a family of domains for which $\lambda_1(D_{\alpha})$ crosses the value $N-1$ as $\alpha$ varies in $(0,\alpha^*)$ in a strict monotonic way. This is not easy, since the behavior of Neumann eigenvalues under domain perturbation can be quite involved, already in the Euclidean case; moreover, Neumann eigenvalues do not enjoy domain monotonicity and futhermore, on the sphere, we don't have the notion of homothety (hence we don't have simple transformations which allow to track explicitly the eigenvalues as in $\mathbb R^N$). In Section \ref{sec2} we construct a family of diffeomorphism $\Phi_\alpha$ on the hemisphere $\mathbb{S}^{N-1}_+$, which are, essentially, dilations with respect to the north pole. Then for the corresponding domain $D_\alpha=\Phi_\alpha(D)$ ($D$ a fixed domain suitably chosen) we are able to study the asymptotic behavior of $\lambda_1(D_\alpha)$, as $\alpha\to 0$, and, what is more important, to compute the derivative $\frac{d}{d\alpha}\lambda_1(D_\alpha).$

The proof that this derivative is negative at $\alpha=1$ (that is, when $\lambda_1(D_1)=N-1$) is the key point to get the so-called transversality condition in the application of Crandall-Rabinowitz theorem.

We also provide an example of a family of domains $D_\alpha$ for which $\lambda_1(D_\alpha)$ is simple (in a neighborhood of $\alpha=1$). However we point out that, generically, for almost all domains on $\mathbb{S}^{N-1}$ all Neumann eigenvalues are simple (see \cite{Henri}).

The other steps of the proof of Theorem \ref{mainth} follow by studying problem \eqref{printro} in a suitable space where, in particular, all functions are invariant by the Kelvin transform. This is inspired by the study of other critical exponent problems (see \cite{GGT}).

Finally we show that the bifurcation is global and that the Rabinowitz alternative holds. We refer to Section \ref{sec3} for the precise statement.
\bigskip

\textbf{Organization of the paper.} The paper is organized as follows. Section \ref{sec2} is devoted to the geometrical question of finding domains on the sphere to which the Crandall-Rabinowitz bifurcation Theorem can be applied. In Section \ref{sec3} we prove the bifurcation results.

The post-theorem text contains a proof sketch for Theorem~\ref{thvecchio}: it “relies on a careful analysis of the Morse index of the standard bubble $U$,” showing that $U$ “becomes an unstable critical point of \eqref{Sobqintro} as soon as $\lambda_1(D)$ crosses the value $N-1$” (cf. Theorem 4.3 in \cite{CPP}). By contrast, “when $\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q$, i.e. it is a strict local minimum.” The analysis also identifies degeneracy at the threshold: “varying the domains $D\subset \mathbb{S}^{N-1}$ … the bubble $U$ is degenerate for any cone $\Sigma_{D_1}$ such that $\lambda_1(D_1)=N-1$,” which “suggests that a bifurcation from the standard bubble could appear when $\lambda_1(D_1)=N-1$,” i.e. “a branch of nonradial solutions … could emanate from the bubble $U$ in $D_1$,” yielding nonradial minimizers/solutions in the regime $\lambda_1(D)<N-1$.

The post-theorem text contains a proof sketch for the preceding theorem: it relies on a careful analysis of the Morse index of the standard bubble $U$, showing that $U$ becomes an unstable critical point of
\beq\label{Sobqintro}
Q_D(u)=\frac{\big(\int_{\Sigma_D}|\nabla u|^2\big)^\frac{1}{2}}{\big(\int_{\Sigma_D} |u|^{2^*}\big)^\frac{1}{2^*}},\qquad u\in\mathcal{D}^{1,2}(\Sigma_D),u\ne 0,
\eeq
as soon as $\lambda_1(D)$ crosses the value $N-1$ (cf. Theorem 4.3 in Colasuonno–Pacella–Pistoia, \emph{Title not provided} (year not provided)). By contrast, when $\lambda_1(D)>N-1$ the bubble $U$ is a stable critical point for $Q_D$, i.e. it is a strict local minimum. The analysis also identifies degeneracy at the threshold: varying the domains $D\subset \mathbb{S}^{N-1}$, the bubble $U$ is degenerate for any cone $\Sigma_{D_1}$ such that $\lambda_1(D_1)=N-1$. This suggests that a bifurcation from the standard bubble could appear when $\lambda_1(D_1)=N-1$, i.e. a branch of nonradial solutions could emanate from the bubble $U$ in $D_1$, yielding nonradial minimizers/solutions in the regime $\lambda_1(D)<N-1.

\label{thvecchio}Chou, Chu, and Pei, “CPP”
    Let $\Sigma_D$ be a cone such that $\bar D\subset \mathbb{S}^{N-1}_+$ and $\lambda_1(D)<N-1$, where $\mathbb{S}^{N-1}_+$ is the half-sphere. Then the minimizers of the Sobolev quotient are nonradial and hence there exist nonradial solutions $v$ of
\label{printro}
\begin{cases}
    -\Delta u=u^{2^*-1}&\text{in }\Sigma_D,\\
    \frac{\partial u}{\partial\nu}=0&\text{on }\partial\Sigma_D\setminus\{0\},\\
    u>0&\text{in }\Sigma_D,
\end{cases}
\ee
which are also fast decaying, i.e.
    $$
    v(x)=O(|x|^{2-N}) \quad \text{as }|x|\to\infty.
    $$

Question:
Let \(N\ge 3\), let \(D\) be a smooth domain of the unit sphere \(\mathbb S^{N-1}\), and let
\[
\Sigma_D=\{x\in\mathbb R^N:x=\rho p,\ p\in D,\ \rho>0\}
\]
be the cone spanned by \(D\). Assume that \(\overline D\subset \mathbb S^{N-1}_+\), where \(\mathbb S^{N-1}_+\) is the upper half-sphere, and that \(\lambda_1(D)<N-1\), where \(\lambda_1(D)\) denotes the first nonzero eigenvalue of \(-\Delta_{\mathbb S^{N-1}}\) on \(D\) with Neumann boundary condition. Consider the Sobolev quotient
\[
Q_D(u)=\frac{\left(\int_{\Sigma_D}|\nabla u|^2\right)^{1/2}}{\left(\int_{\Sigma_D}|u|^{2^*}\right)^{1/2^*}},
\qquad 2^*=\frac{2N}{N-2},
\]
for \(u\in \mathcal D^{1,2}(\Sigma_D)\setminus\{0\}\), where
\[
\mathcal D^{1,2}(\Sigma_D)=\{u\in L^{2^*}(\Sigma_D): |\nabla u|\in L^2(\Sigma_D)\}.
\]
Here "nonradial" means not depending only on \(|x|\). Which statement holds about minimizers of \(Q_D\) and the corresponding critical Neumann problem
\[
\begin{cases}
-\Delta u=u^{2^*-1} & \text{in } \Sigma_D,\\
\dfrac{\partial u}{\partial \nu}=0 & \text{on } \partial\Sigma_D\setminus\{0\},\\
u>0 & \text{in } \Sigma_D?
\end{cases}
\]

Every minimizer of \(Q_D\) is nonradial. Consequently, there exists a nonradial positive solution \(v\in \mathcal D^{1,2}(\Sigma_D)\) of
\[
\begin{cases}
-\Delta v=v^{2^*-1} & \text{in } \Sigma_D,\\
\dfrac{\partial v}{\partial \nu}=0 & \text{on } \partial\Sigma_D\setminus\{0\},\\
v>0 & \text{in } \Sigma_D,
\end{cases}
\]
and this solution is fast decaying in the sense that
\[
v(x)=O(|x|^{2-N})\qquad\text{as }|x|\to\infty.
\]

Choices:
B.
Every minimizer of \(Q_D\) is radial. Consequently, every positive solution arising from a minimizer coincides, up to the natural rescaling of the standard bubble, with a trivial radial solution in \(\Sigma_D\), and in particular satisfies
\[
v(x)=O(|x|^{2-N})\qquad\text{as }|x|\to\infty.
\]
C.
There exists at least one nonradial positive solution \(v\in \mathcal D^{1,2}(\Sigma_D)\) of
\[
\begin{cases}
-\Delta v=v^{2^*-1} & \text{in } \Sigma_D,\\
\dfrac{\partial v}{\partial \nu}=0 & \text{on } \partial\Sigma_D\setminus\{0\},\\
v>0 & \text{in } \Sigma_D,
\end{cases}
\]
which is obtained from a minimizer of \(Q_D\).
D.
Every minimizer of \(Q_D\) is nonradial if and only if \(\lambda_1(D)\le N-1\). Consequently, for every smooth domain \(D\) with \(\overline D\subset \mathbb S^{N-1}_+\) and \(\lambda_1(D)\le N-1\), there exists a nonradial positive solution \(v\in \mathcal D^{1,2}(\Sigma_D)\) of
\[
\begin{cases}
-\Delta v=v^{2^*-1} & \text{in } \Sigma_D,\\
\dfrac{\partial v}{\partial \nu}=0 & \text{on } \partial\Sigma_D\setminus\{0\},\\
v>0 & \text{in } \Sigma_D,
\end{cases}
\]
and such a solution is fast decaying:
\[
v(x)=O(|x|^{2-N})\qquad\text{as }|x|\to\infty.
\]
E.
There exists a constant \(\varepsilon>0\), independent of the particular smooth domain \(D\subset \mathbb S^{N-1}_+\) with \(\lambda_1(D)<N-1\), such that every minimizer of \(Q_D\) is nonradial and every corresponding positive solution \(v\) of the critical Neumann problem satisfies the uniform decay estimate
\[
|v(x)|\le \varepsilon |x|^{2-N}\qquad\text{for all sufficiently large }|x|.
\]


\label{theorem: bound rango}
For every $k\geq 2$ and every $d_1,\dots,d_k\geq 3$ we have
$$
\rk_{\mathbf{d}}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq  2^{k-1}\left(d_1+\cdots+d_k-2k+2\right).
$$

\begin{theorem}\label{theorem: bound rango}
For every $k\geq 2$ and every $d_1,\dots,d_k\geq 3$ we have
$$
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq  2^{k-1}\left(d_1+\cdots+d_k-2k+2\right).
$$
\end{theorem}

\label{subsection: commenti sul bound}

As already emphasized, the general idea of bounding the rank of tensor product of $W$-states by looking at the containment of $W_{d_1}\otimes \cdots \otimes W_{

\begin{proposition}\label{prop: rango TJ}
Let $k\geq 2$, let $\bfd\in \NN_{\geq 3}^k$ and let $d=d_1+\cdots +d_k$. Let $\mathcal{F}\subseteq \CC[u,v]_d$ be the family of degree $d$-homogenous polynomial defined as
$$
\mathcal{F}= \left\{  F=\sum_{i=0}^{k-2}a_i\binom{d}{i}u^{d-i}v^i+\binom{d}{d-k}a_{d-k}u^{k}v^{d-k}+\sum_{i=d-k+2}^d a_i \binom{d}{i}u^{d-i}v^i, \text{ for }a_i\in \CC \text{ with } a_{d-k}\neq 0 \right\}.
$$
Then
$$
\min_{F\in\mathcal F}\rk F=d-2k+2.
$$
In particular, if  $2k\leq\left\lfloor \frac d2\right\rfloor +1$ the minimum is achieved by any $F\in\mathcal F$ with $a_{k-2}\neq 0$. Otherwise, the minimum is achieved by any $F\in \mathcal{F}$ with $a_{k-2}\neq 0$ and $a_i=0$ for $i\in \{0,\dots,k-3\} \cup \{d-k+2,\dots,d \}$.
\end{proposition}

Computing the rank of a tensor is generally NP-hard, see \cite{Has90, rankNPhard}. For this reason, one often focuses on special families of tensors that serve as meaningful benchmarks either because they originate from applications, or because they exhibit particularly interesting behavior. One such family is the tensor product of generalized $W$-state: $$W_{d_1}\otimes \cdots \otimes W_{d_k}\in \underbrace{\mathbb{C}^2\otimes \cdots \otimes \mathbb{C}^2}_{d_1+\cdots +d_k-\text{times}}
$$
where each $W$-state $W_d$ is defined as
$$
W_d=\sum_{i=1}^d e^{(1)}_{1}\otimes \cdots \otimes e^{(i-1)}_1\otimes e^{(i)}_2 \otimes e^{(i+1)}_1\otimes \cdots \otimes e^{(d)}_1 \in (\mathbb{C}^2)^{\otimes d}, \quad \spann\{e^{(i)}_1,e^{(i)}_2 \}\cong \mathbb{C}^2.
$$
Focusing for a moment on a single copy, the generalized $W$-state itself is a highly interesting tensor. It is a symmetric tensor and one of the simplest examples illustrating the non-semicontinuity of tensor rank. Moreover, the expression of the general element of the tangential variety $\tau(\nu((\mathbb{P}^1)^{\times d}))$ of the Segre image $\nu((\mathbb{P}^1)^{\times d})$ is $W_d$. This is because actually $W_d\in \langle \nu(Z)\rangle$, where $Z\subset (\mathbb{P}^1)^{\times d}$ is a zero-dimensional scheme of length 2 supported at $\otimes_{i=1}^d e^{(i)}_{1} $, called a 2-jet \cite{alexander1997interpolation}, so actually $W_d$ lies on a line that is tangent to $\nu((\mathbb{P}^1)^{\times d})$ at $\otimes_{i=1}^d e^{(i)}_{1} $. The $W$-state is a rank-$d$ tensor and it is an example of tensor for which tensor rank and symmetric tensor rank coincide \cite{rankTangentialBB}. It has infinitely many decompositions computing its rank and in particular, one can choose any elementary tensor except for $\otimes_{i=1}^d e^{(i)}_{1}$ to be part of one of its minimal (symmetric) rank decomposition \cite{carlini2017waring, BOS}.

From the point of view of complexity theory, $W$-states are related to the so-called Coppersmith-Winogard tensors \cite{coppersmith1987matrix}, which are tensors of interest in the study of the matrix multiplication tensor. In particular, $W$-states are the outer structure of many tensors, including Strassen tensor and truncated polynomial multiplication, used in the so-called \emph{laser method}, a strategy introduced by V. Strassen in \cite{Str86} for computing the complexity of matrix multiplication.

In the three-factors case $W_3$ plays a historical central role in quantum information theory, being one of the two fundamental classes of genuine tripartite entanglement \cite{dur2000three}, and this is where the name \emph{$W$-state} comes from. This makes the $W$-state a structurally very rich object at the intersection of geometry, algebraic complexity, and quantum information.

Given how special $W_d$ is, it is natural to expect that the tensor product of several copies, $W_{d_1}\otimes \cdots \otimes W_{d_k} $, is also quite special. This expectation is reinforced for instance by the role that the tensor product of two copies of $W$-states had in the study of the multiplicativity of tensor rank under tensor product: indeed, \cite{CJZ} proved that the rank of $W_3\otimes W_3$ is strictly less than the naive multiplicative guess $3\cdot 3$. This was the first explicit example showing that the rank of the tensor product of two tensors is not the product of the two ranks. Shortly after, \cite{CF18} proved that the rank of $W_3\otimes W_3$ is 8. A systematic approach to the strict submulticativity property can be found in \cite{BBGOV}, while, motivated by applications in complexity theory and quantum information theory, the submultiplicativity of the Kronecker tensor product of many copies of $W$-states has been studied for instance in \cite{chen2010tensor, zuiddam2017note}. Some Kronecker products of $W$-states show also other interesting connections. For instance, $W_3\boxtimes W_3$ is a symmetric tensor corresponding to the cubic hypersurface of $\mathbb{P}^3$ made of a quadric and a tangent hyperplane and, by \cite{Seg42}, this is the only quaternary cubic having maximal rank 7. As a second example, a particular projection of $W_3\boxtimes W_3\boxtimes W_3$ is the so-called Bjorklund-Kaski tensor, which appears in \cite{BK24} and is used for proving that the asymptotic rank conjecture \cite{Str94} and the set cover conjecture \cite{CFKLMMPS15, KT19} cannot be both true.

Hence, determining the rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$ becomes a particularly meaningful problem. \cite{BBCG} gave the first bounds on the rank of the tensor product of an arbitrary number of $W$-states
 and these remain the best known results to date, with the notable exception of the case $k=2$ where the bound was later improved in \cite[Theorem 1.8(i)]{gal}.

In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\mathbb{C}^{2})^{\otimes d_i}$. We remark that $W_{d_1}\otimes \cdots \otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\in \mathop{\rm Sym}\nolimits^{d_1}\mathbb{C}^{n_1}\otimes \cdots \otimes \mathop{\rm Sym}\nolimits^{d_k}\mathbb{C}^{n_k}$ the partially symmetric rank of $T$, denoted as $\rk_{\mathbf{d}}(T)$, is the minimum integer $r$ such that $T=\sum_{i=1}^r (v^{(1)}_i)^{\otimes d_1}\otimes \cdots \otimes (v^{(k)}_i)^{\otimes d_k}$, where $v^{(j)}_i\in \mathbb{C}^{n_j}$. Our main result is the following.

In this paper, we make progress along this direction by providing a new sharp upper bound on the rank of tensor products of generalized $W$-states $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\CC^{2})^{\otimes d_i}$. We remark that $W_{d_1}\otimes \cdots \otimes W_{d_k}$ is a partially symmetric tensor and for a partially symmetric tensor $T\in \sym^{d_1}\CC^{n_1}\otimes \cdots \otimes \sym^{d_k}\CC^{n_k}$ the partially symmetric rank of $T$, denoted as $\rk_{\bfd}(T)$, is the minimum integer $r$ such that $T=\sum_{i=1}^r (v^{(1)}_i)^{\otimes d_1}\otimes \cdots \otimes (v^{(k)}_i)^{\otimes d_k}$, where $v^{(j)}_i\in \CC^{n_j}$. Our main result is the following.

In addition to providing a theoretical upper bound on the tensor rank, \Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$; see the forthcoming \Cref{remark: commento algo}.

As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \begin{corollary}\label{corollary: bound rango vero}
Let $k\geq 2$, fix positive integers $d_i\geq 3$ for $i=1,\dots,k$.
The rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\CC^{2})^{\otimes d_i}$ is at most $2^{k-1}(d_1+\dots+d_k-2k+2)$.
\end{corollary}

We have now all the tools to prove our main \Cref{theorem: bound rango}, which we restate below for convenience.
\begin{theorem*}(\Cref{theorem: bound rango}) Let $k\geq 2$, $\bfd\in \NN_{\geq 3}^k$ and $d=d_1+\cdots +d_k$. Then
$$
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq 2^{k-1}(d-2k+2).
$$
\end{theorem*}
\begin{proof}
By \Cref{remark: W-state sta nel quadrello} $W_{d_1}\otimes\cdots \otimes W_{d_k}$ belongs to $\spann\left\{ \cup_{J\subseteq [2,k]} \mathcal{C}_{\bfd}^J\right\} $, so we have
$$
\prod_{j=1}^kd_j W_{d_1}\otimes\cdots \otimes W_{d_k}=\sum_{J\subseteq [2,k]}T^J, \,\hbox{ for some }T^J\in\spann\mathcal{C}_{\bfd}^J.
$$
Writing $T^J$ with respect to the basis of \Cref{remark: multi hom binary in forma binaria emplice}, we have
$$T^J=\sum_{s=0}^{d}\alpha_s^J\left(\sum_{\bfi\in\mathcal A_{\bfd,s}}\varepsilon_\bfi^Jx_{1,0}^{i_1}x_{1,1}^{d_1-i_1}\otimes\cdots\otimes x_{k,0}^{i_k}x_{k,1}^{d_k-i_k}\right),$$
where the $\alpha^J_s$'s are a solution of the linear system $\mathcal S$ of \Cref{prop: mortale}.
Using the isomorphism $\varphi^J$ defined in \Cref{remark: multi hom binary in forma binaria emplice}, each $T^J$ can be seen as a binary form of degree $d$ in $\spann\mathcal C_\bfd^J$. More precisely, we have
$$\varphi^J(T^J)=\sum_{s=0}^d\alpha_s^J\binom{d}{s}u^{d-s}v^s.$$
It follows that $\rk_\bfd T^J\leq\rk\varphi^J(T^J)$, where $\rk\varphi^J(T^J)$ is the Waring rank of the binary form $\varphi^J(T^J)$ and, by \Cref{remark: multi hom binary in forma binaria emplice}, it is equal to the $\mathcal C_\bfd^J$-rank of $T^J$.
By \Cref{prop: mortale}, we have $\alpha_s^J=0$ for any $s\in[k-1,d-k-1]\cup\left\{d-k+1\right\}$ and $\alpha^J_{d-k}\neq 0$. Moreover, whether $2k\leq\left\lfloor\frac{d}{2}\right\rfloor+1$ or $2k>\left\lfloor\frac{d}{2}\right\rfloor+1$, by \Cref{prop: mortale} we can choose the $\alpha_s^J$'s such that $\varphi^J(T^J)$ is a minimizer as in \Cref{prop: rango TJ}, so that $\rk (\varphi^J(T^J))=d-2k+2$. Hence, we have
\begin{equation*}
\rk_{\bfd}(W_{d_1}\otimes \cdots \otimes W_{d_k})\leq\sum_{J\subseteq[2,k]}\rk_\bfd T^J\leq \sum_{J\subseteq[2,k]}\rk \varphi^J(T^J)= 2^{k-1}(d-2k+2).\qedhere
\end{equation*}
\end{proof}
Since for a tensor $T\in \sym^{d_1}\CC^2\otimes \cdots \otimes \sym^{d_k}\CC^2$ we have $\rk(T)\leq \rk_{\bf d}(T)$, the previous result gives a bound also on the tensor rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$.

\Cref{example: step4}, together with \Cref{example: step1}, \Cref{example: step2} and \Cref{example: step3}, concretely shows that, following our procedure, it is possible to construct an explicit decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$ having length as in the bound of \Cref{theorem: bound rango} as we now explain in detail.
\begin{remark}\label{remark: commento algo}
Let $\bfd\in\NN^k_{\geq 3}$. For $s\in[0,d]\setminus\{d-k\}$ we choose one $\bf i \in \A_{\bf d,s }$ and set $\varepsilon^J_{\bf i}=1$ for any $J\subseteq[2,k]$, and for $s=d-k$ and $\bfi=(d_1-1,\dots,d_k-1)$ we set $\varepsilon_\bfi^J=1$ for any $J\subseteq[2,k]$. In order to find a decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$ following our procedure, we have to first find $T^J\in\spann(\mathcal C_{\bfd}^J)$ such that
$$\prod_{j=1}^kd_jW_{d_1}\otimes\cdots\otimes W_{d_k}=\sum_{J\subseteq[2,k]}T^J.$$
The coordinates of the $T^J$'s are of the form
$$T^J=(\varepsilon_{\bfi}^J\alpha_s^J)_{\substack{0\leq s\leq d\\ \bfi\in\A_{\bfd,s}}}$$
and, by \Cref{prop: rango TJ}, it is enough to take $\alpha_{k-2}^J\neq0$ and $\alpha_{d-k}^J\neq 0$ for every $J$ to minimize the length of the decomposition. The easiest possible choice is given by \Cref{prop: mortale}: we take $\alpha_s=0$ for $s\in[0,d]\setminus\{k-2,d-k\}$, $\alpha_{k-2}=(-1)^{|J|}$ and $\alpha_{d-k}=2^{k-1}$. Hence, when we compute $\varphi^J(T^J)$ we only have two possibilities:
$$\varphi^J(T^J)=\begin{cases}
 \binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\binom{d}{d-k}u^kv^{d-k},&\text{ if $|J|$ is even}\\
 -\binom{d}{k-2}u^{d-k+2}v^{k-2}+2^{k-1}\binom{d}{d-k}u^kv^{d-k},&\text{ otherwise.}
\end{cases}$$
As a consequence, it is enough to apply the Sylvester's algorithm to only two binary forms. Once a minimal decomposition of the two possible $\varphi^J(T^J)$'s is found it is enough to apply $(\varphi^J)^{-1}$ for all the $J\subseteq[2,k]$ to find a partially symmetric decomposition of the product of $W$-states. Note that this drastically reduce the computational cost. Indeed, the decompositions of the two binary forms and the application of $(\varphi^J)^{-1}$ are computationally cheap. The core of the advantage of this method is that the combinatorics of the $\varepsilon_\bfi^J$'s allows to avoid many computations. We summarized the procedure in \Cref{algo}.
\end{remark}

\begin{theorem}\label{theorem: bound rango bordo}
Let $k\geq 2$, let $\bfd\in \NN_{\geq 3}^k$ and let $d=d_1+\cdots+d_k$. Then
$$
\underline{\rk}_\bfd(W_{d_1}\otimes\cdots\otimes W_{d_k})= 2^{k}.
$$
\end{theorem}
\begin{proof}
We use the catalecticant bound for border rank of \cite[Corollary 5.5]{gal}. In order to do that we have to produce a partially symmetric flattening with rank at least $2^k$. Let us consider the following flattening of $W_{d_1}\otimes \cdots \otimes W_{d_k}$:
\[
F : \sym^2(\mathbb{C}^2) \otimes \dots \otimes \sym^2(\mathbb{C}^2) \rightarrow
\sym^{d_1-2}(\mathbb{C}^2)^* \otimes \dots \otimes \sym^{d_k-2}(\mathbb{C}^2)^*.
\]
Let $\mathbb{C}[\alpha_{1,0},\alpha_{1,1},\dots,\alpha_{k,0},\alpha_{k,1}]$ be the ring of multigraded derivations on $\CC[x_{1,0},x_{1,1},\dots,x_{k,0},x_{k,1}]$ with $\alpha_{i,j}$ defined as the dual of $x_{i,j}$. By apolarity theory, the rank of the linear map $F$ is exactly the dimension of the degree $(2,\dots,2)$ component of the quotient of the multigraded ring $\mathbb{C}[\alpha_{1,0},\alpha_{1,1},\dots,\alpha_{k,0},\alpha_{k,1}]$ by the ideal $I_{d}$ apolar to the W-state, i.e. $I_{d}=(\alpha_{1,0}^2,\dots,\alpha_{k,0}^2,\alpha_{1,1}^{d_1},\dots,\alpha_{k,1}^{d_k})$. For $\varepsilon_i \in \{0,1\}$ the $2^k$ forms
$$F_{\varepsilon_1,\dots,\varepsilon_{k}}=\alpha_{1,0}^{\varepsilon_1}\alpha_{1,1}^{2-\varepsilon_1} \dots \alpha_{k,0}^{\varepsilon_k}\alpha_{k,1}^{2-\varepsilon_k}$$ are linear independent of multidegree $(2,\dots, 2)$, with $F_{\varepsilon_1,\dots,\varepsilon_{k}} \notin I_d$. This concludes the proof.
\end{proof}

\Cref{theorem: bound rango} improves \cite[Theorem 3.6]{BBCG} by $2^k(k-1)$. For $k=2$ it coincides with the bound provided in \cite[Theorem 1.8(i)]{gal} and when all $d_i=3$ it matches the bound of \cite[Theorem 3.3]{BBCG}. The bound of \Cref{theorem: bound rango} is sharp as for $k=2$ and $d_1=d_2=3$ it gives rank 8.

 In addition to providing a theoretical upper bound on the tensor rank, \Cref{theorem: bound rango} is constructive. Indeed, the proof yields an explicit decomposition of length equal to the bound that can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}). This decreases a lot the computational cost for computing a partially symmetric decomposition of $W_{d_1}\otimes\cdots\otimes W_{d_k}$; see the forthcoming \Cref{remark: commento algo}.

 As an immediate consequence the same bound holds also for the rank of tensor product of $W$-states. \begin{corollary}\label{corollary: bound rango vero}
Let $k\geq 2$, fix positive integers $d_i\geq 3$ for $i=1,\dots,k$.
The rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}\in \bigotimes_{i=1}^k(\mathbb{C}^{2})^{\otimes d_i}$ is at most $2^{k-1}(d_1+\dots+d_k-2k+2)$.
\end{corollary}

Our methods to achieve this result are geometric and, as in \cite[Theorem 3.6]{BBCG}, they rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.

\subsection*{Outline of the paper}  We set up our notation and give standard preliminaries in \Cref{section: preliminaries}, while \Cref{section: solving the system} contains the main technical results to get our bounds. We start \Cref{section: computing the bounds} by briefly reviewing the theory of catalecticants needed to compute the rank of a family of binary homogeneous polynomials (\Cref{prop: rango TJ}). Then, we prove our main \Cref{theorem: bound rango} and in \Cref{algo} we give a recipe to explicitly compute a partially symmetric decomposition of length given by our bound. \Cref{theorem: bound rango bordo} computes the border partially symmetric rank of $W_{d_1}\otimes \cdots \otimes W_{d_k}$. We conclude with a detailed comment on the method in \Cref{subsection: commenti sul bound}.

\subsection*{Acknowledgements}
The authors are grateful to Fulvio Gesmundo, Alessandro Gimigliano and Joachim Jelisiejew for interesting conversations on the topic.

S. Canino has been funded by the Italian Ministry of
University and Research in the framework of the Call for Proposals for
scrolling of final rankings of the PRIN 2022 call - Protocol no.
2022NBN7TL.
A. Casarotti has been funded by the European Union under the project NextGenerationEU. PRIN 2022, CUP: F53D23002600006.
P. Santarsiero was supported by the European Union under NextGenerationEU. PRIN 2022, Prot. 2022E2Z4AK and PRIN 2022 SC-CUP: I53C24002240006.

The post-theorem text does not give a step-by-step proof sketch for \Cref{theorem: bound rango}, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see \Cref{algo}).”

The post-theorem text does not give a step-by-step proof sketch for the main theorem, but it indicates the proof’s nature and main idea: the result is obtained by “geometric” methods and “rely on finding a curve in $\mathcal{C}\subset (\mathbb{P}^1)^{\times k}$ such that the span of the image of $\mathcal{C}$ under the Segre-Veronese embedding contains $W_{d_1}\otimes \cdots \otimes W_{d_k}$.” It also states the proof is “constructive,” yielding “an explicit decomposition of length equal to the bound” that “can be achieved by computing the rank decomposition of essentially one binary form (see
\label{algo}
\end{algorithm}
\subsection{Further remarks on \Cref{theorem: bound rango} }\label{subsection: commenti sul bound}

As already emphasized, the general idea of bounding the rank of tensor).”

Question:
Let \(k\ge 2\) and let \(d_1,\dots,d_k\ge 3\) be integers, and write \(\mathbf d=(d_1,\dots,d_k)\). For each \(d\), define the generalized \(W\)-state
\[
W_d=\sum_{i=1}^d e^{(1)}_1\otimes\cdots\otimes e^{(i-1)}_1\otimes e^{(i)}_2\otimes e^{(i+1)}_1\otimes\cdots\otimes e^{(d)}_1\in (\mathbb C^2)^{\otimes d},
\]
where each \(\operatorname{span}\{e^{(i)}_1,e^{(i)}_2\}\cong \mathbb C^2\). View
\(W_{d_1}\otimes\cdots\otimes W_{d_k}\) as an element of
\(\operatorname{Sym}^{d_1}(\mathbb C^2)\otimes\cdots\otimes \operatorname{Sym}^{d_k}(\mathbb C^2)\), and let \(\operatorname{rk}_{\mathbf d}\) denote its partially symmetric rank, i.e. the minimum number of summands of the form \(\ell_1^{d_1}\otimes\cdots\otimes \ell_k^{d_k}\) needed to express it. Which statement holds for every such choice of \(k,d_1,\dots,d_k\)?

\[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]

Choices:
B. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]
C. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}(d_1+\cdots+d_k).\]
D. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})\le 2^{k-1}\bigl(d_1\cdots d_k-2k+2\bigr).\]
E. \[\operatorname{rk}_{\mathbf d}(W_{d_1}\otimes\cdots\otimes W_{d_k})= 2^{k-1}\bigl(d_1+\cdots+d_k-2k+2\bigr).\]