Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

1.   Introduction

We consider the Neumann problem

in the unbounded cone $\Sigma_\omega: = \{tx:x\in\omega\text{ and }t > 0\}$, where $\omega$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$, and $2^*: = \frac{2N}{N-2}$ is the critical Sobolev exponent.

It is well known that, if $\omega = \mathbb{S}^{N-1}$, i.e., if $\Sigma_\omega$ is the whole space $\mathbb{R}^{N}$, then the only positive solutions to the critical problem

are the rescalings and translations of the standard bubble $U$ defined in (2.3). Moreover, they are the only nontrivial radial solutions to (1.2), up to sign. It is immediately deduced that, up to sign, the restriction of the bubbles (3.1) to $\Sigma_\omega$ are the only nontrivial radial solutions of (1.1) in any cone; see Proposition 3.4. In addition, if the cone $\Sigma_\omega$ is convex, it was shown in [8,Theorem 2.4] that these are the only positive solutions to (1.1). The convexity property of the cone is crucial in the proof of this result, and it is strongly related to a relative isoperimetric inequality obtained in ^[7].

The aim of this paper is to establish the existence of nonradial solutions to (1.1), both positive and sign-changing. As mentioned above, the positive ones can only exist in nonconvex cones. On the other hand, nodal radial solutions to (1.1) do not exist, as this would imply the existence of a nontrivial solution to problem (2.5) in the bounded cone $\Lambda_\omega: = \{tx:x\in\omega\text{ and }t\in(0, 1)\}$, which is impossible because of the Pohozhaev identity (2.6) and the unique continuation principle.

For the problem (1.2) in $\mathbb{R}^N$ various types of sign-changing solutions are known to exist; see ^[2,3,4,5]. In particular, a family of entire nodal solutions, which are invariant under certain groups of linear isometries of $\mathbb{R}^N$, were exhibited in ^[2]. These solutions arise as blow-up profiles of symmetric minimizing sequences for the critical equation in a ball, and are obtained through a fine analysis of the concentration behavior of such sequences.

Here we use some ideas from ^[2] to produce sign-changing solutions to (1.1), but we exploit a different kind of symmetry. Our main result shows that, if $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$ and if the cone $\Sigma_\omega$ has a point of convexity in the sense of Definition 2.6, then the problem (1.1) has an axially antisymmetric least energy solution, which is nonradial and changes sign; see Theorem 2.8. As far as we know, this is the first existence result of a nodal solution to (1.1).

Next, we investigate the existence of positive nonradial solutions. In this case we do not require the cone to have any particular symmetry. We establish the existence of a positive nonradial solution to (1.1) under some conditions involving the local convexity of $\Sigma_\omega$ at a boundary point and the measure of the bounded cone $\Lambda_\omega$; see Corollary 3.5 and Theorem 3.6. We refer to Section 3 for the precise statements and further remarks.

2.   A nonradial sign-changing solution

If $\Omega$ is a domain in $\mathbb{R}^N$ we consider the Sobolev space

with the norm

We denote by $J_\Omega:D^{1, 2}(\Omega)\to\mathbb{R}$ the functional given by

and its Nehari manifold by

For $u\in D^{1, 2}(\Omega)\backslash \{0\}$ let $t_u\in (0, \infty)$ be such that $t_uu\in \mathcal{N}(\Omega)$. Then,

Hence,

We set $c_\infty: = c_{\mathbb{R}^N}$. It is well known that this infimum is attained at the function

which is called the standard bubble, and at every rescaling and translation of it, and that

where $S$ is the best constant for the Sobolev embedding $D^{1, 2}(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ and let $\omega$ be a smooth domain in $\mathbb{S}^{N-1}$ with nonempty boundary, i.e., $\omega$ is connected and open in $\mathbb{S}^{N-1}$ and its boundary $\partial\omega$ is a smooth $(N-2)$-dimensional submanifold of $\mathbb{S}^{N-1}$. The nontrivial solutions to the Neumann problem (1.1) in the unbounded cone

are the critical points of $J_{\Sigma_\omega}$ on $\mathcal{N}(\Sigma_\omega)$.

To produce a nonradial sign-changing solution for (1.1) we introduce some symmetries. We write a point in $\mathbb{R}^N$ as $x = (x', x_N)\in\mathbb{R}^{N-1}\times\mathbb{R}$, and consider the reflection $\varrho(x', x_N): = (-x', x_N)$. Then, a subset $X$ of $\mathbb{R}^N$ will be called $\varrho$-invariant if $\varrho x\in X$ for every $x\in X$, and a function $u:X\to\mathbb{R}$ will be called $\varrho$-equivariant if

Note that every nontrivial $\varrho$-equivariant function is nonradial and changes sign.

Throughout this section we will assume that $\omega$ is $\varrho$-invariant. Note that $(0, \pm 1)\not\in\partial\omega$ because $\partial\omega$ is smooth. Hence, $\varrho x\neq x$ for every $x\in\partial\Sigma_\omega \backslash \{0\}$. Our aim is to show that (1.1) has a $\varrho$-equivariant solution. We set

and

Define

and set $\Gamma_1: = \partial\Lambda_{\omega}\backslash \overline{\omega}$. In $\Lambda_\omega$ we consider the mixed boundary value problem

We point out that (2.5) does not have a nontrivial solution. Indeed, by the well known Pohozhaev identity, a solution to (2.5) must satisfy

because $s\cdot\nu = 0$ for every $s\in\Gamma_1$ and $s\cdot\nu = 1$ for every $s\in\omega$. Therefore $\frac{\partial u}{\partial\nu}$ vanishes on $\omega$. But then the trivial extension of $u$ to the infinite cone $\Sigma_\omega$ solves (1.1), contradicting the unique continuation principle.

Let $V(\Lambda_\omega)$ be the space of functions in $D^{1, 2}(\Lambda_{\omega})$ whose trace vanishes on $\omega$. Note that $V(\Lambda_\omega)\subset D^{1, 2}(\Sigma_{\omega})$ via trivial extension. Let $J_{\Lambda_\omega}:V(\Lambda_\omega)\to\mathbb{R}$ be the restriction of $J_{\Sigma_\omega}$ to $V(\Lambda_\omega)$ and set

To produce a sign-changing solution for the problem (1.1) we will study the concentration behavior of $\varrho$-equivariant minimizing sequences for (2.5). We start with the following lemmas.

Lemma 2.1. $0 < c^\varrho_{\Lambda_\omega} = c^\varrho_{\Sigma_\omega} \leq c_\infty.$

Proof. It is shown in [8,Theorem 2.1] that $c^\varrho_{\Lambda_\omega} > 0$.

Since $\mathcal{N}^\varrho(\Lambda_\omega)\subset\mathcal{N}^\varrho(\Sigma_\omega)$, we have that $c^\varrho_{\Lambda_\omega} \geq c^\varrho_{\Sigma_\omega}$. To prove the opposite inequality, let $\varphi_k\in\mathcal{N}^\varrho(\Sigma_\omega)\cap\mathcal{C}^\infty(\overline{\Sigma}_\omega)$ be such that $\varphi_k$ has compact support and $J(\varphi_k)\to c^\varrho_{\Sigma_\omega}$ as $k\to\infty$. Then, we may choose $\varepsilon_k > 0$ such that the support of $\widetilde{\varphi}_k(x) : = \varepsilon_k^{-(N-2)/2} \varphi_k (\varepsilon_k^{-1}x)$ is contained in $\overline{\Lambda}_\omega \backslash \overline{\omega}$. Thus, $\widetilde{\varphi}_k \in \mathcal{N}^\varrho(\Lambda_\omega)$ and, hence,

Letting $k \to \infty$ we conclude that $c^\varrho_{\Lambda_\omega} \leq c^\varrho_{\Sigma_\omega}$.

To prove that $c^\varrho_{\Sigma_\omega} \leq c_\infty$ we fix a point $\xi\in\partial\Sigma_\omega \backslash \{0\}$ and a sequence of positive numbers $\varepsilon_k\to 0$, and we set $\Sigma_k: = \varepsilon_k^{-1}(\Sigma_\omega -\xi)$. Since $\partial\Sigma_\omega \backslash \{0\}$ is smooth, the limit of the sequence of sets $(\Sigma_k)$ is the half-space

where $\nu$ is the exterior unit normal to $\Sigma_\omega$ at $\xi$. Let $u_k(x): = \varepsilon_k^{(2-N)/2}U(\frac{x-\xi}{\varepsilon_k})$, where $U$ is the standard bubble (2.3). Then,

The function

is $\varrho$-equivariant, and from (2.4), (2.8) and (2.9) we obtain

This concludes the proof.

Lemma 2.2. Given a domain $\Omega$ in $\mathbb{R}^N$ and $\varepsilon > 0$, we set $\Omega_{\varepsilon}: = \{\varepsilon^{-1}x:x\in\Omega\}$. If $\partial\Omega$ is Lipschitz continuous, then there exist linear extension operators $P_{\varepsilon}:W^{1, 2}(\Omega_{\varepsilon})\to D^{1, 2}(\mathbb{R}^N)$ and a positive constant $C$, independent of $\varepsilon$, such that

$(i) \; (P_{\varepsilon}\, u)(x) = u(x)$ for every $x\in\Omega_{\varepsilon}$.

$(ii)\; \int_{\mathbb{R}^N}|\nabla(P_{\varepsilon}\, u)|^2\leq C\int_{\Omega_{\varepsilon}}|\nabla u|^2$.

$(iii)\; \int_{\mathbb{R}^N}|P_{\varepsilon}\, u|^{2^*}\leq C\int_{\Omega_{\varepsilon}}|u|^{2^*}.$

$(iv)$ If $\Omega$ is $\varrho$-invariant, then $P_{\varepsilon}\, u$ is $\varrho$-equivariant if $u$ is $\varrho$-equivariant.

Proof. The existence of an extension operator $P_{\varepsilon}:W^{1, 2}(\Omega_{\varepsilon})\to D^{1, 2}(\mathbb{R}^N)$ satisfying $(i)-(iii)$ is well known, and the fact that the constant $C$ does not depend on $\varepsilon$ was proved in [6,Lemma 2.1]. To obtain $(iv)$ we replace $P_{\varepsilon}u$ by the function $x\mapsto\frac{1}{2}[(P_{\varepsilon}\, u)(x)-(P_{\varepsilon}\, u)(\varrho x)]$.

The following proposition describes the behavior of minimizing sequences for $J_{\Lambda_\omega}$ on $\mathcal{N}^\varrho(\Lambda_\omega)$.

Proposition 2.3. Let $u_k\in\mathcal{N}^\varrho(\Lambda_\omega)$ be such that

Then, after passing to a subsequence, one of the following statements holds true:

$(i)$ There exist a sequence of positive numbers $(\varepsilon_k)$, a sequence of points $(\xi_k)$ in $\Gamma_1$ and a function $w\in D^{1, 2}(\mathbb{R}^N)$ such that $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \bar{\omega}\cup\{0\})\to\infty$, $w|_\mathbb{H}$ solves the Neumann problem

in some half-space $\mathbb{H}$, $J_{\mathbb{H}}(w) = \frac{1}{2}c_\infty$,

and $c^\varrho_{\Sigma_\omega} = c^\varrho_{\Lambda_\omega} = c_\infty$.

$(ii)$ There exist a sequence of positive numbers $(\varepsilon_k)$ with $\varepsilon_k\to 0$, and a $\varrho$-equivariant solution $w\in D^{1, 2}(\Sigma_\omega)$ to the problem (1.1) such that

and $J_{\Sigma_\omega}(w) = c^\varrho_{\Sigma_\omega} = c^\varrho_{\Lambda_\omega}\leq c_\infty$.

Proof. Since

the sequence $(u_k)$ is bounded and, after passing to a subsequence, $u_k\rightharpoonup u$ weakly in $V(\Lambda_{\omega})$. Then, $J'_{\Lambda_\omega}(u) = 0$. Since the problem (2.5) does not have a nontrivial solution, we conclude that $u = 0$.

Fix $\delta \in (0, \frac{N}{2}c^\varrho_{\Lambda_\omega})$. As

there are bounded sequences $(\varepsilon_k)$ in $(0, \infty)$ and $(x_k)$ in $\mathbb{R}^N$ such that, after passing to a subsequence,

where $B_r(x): = \{y\in \mathbb{R}^N:|y-x| < r\}$. Note that, as $\delta > 0$, we have that $\mathrm{dist}(x_k, \Lambda_\omega) < \varepsilon_k$. We claim that, after passing to a subsequence, there exist $\xi_k\in\bar{\Lambda}_\omega$ and $C_0 > 0$ such that

and one of the following statements holds true:

$(a) \; \xi_k = 0$ for all $k\in\mathbb{N}$.

$(b) \; \xi_k\in\partial\omega = \overline{\omega}\cap\overline{\Gamma}_1$ for all $k\in\mathbb{N}$.

$(c) \; \xi_k\in\Gamma_1$ for all $k\in\mathbb{N}$ and $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \bar{\omega}\cup\{0\})\to \infty$.

$(d) \; \xi_k\in\omega$ for all $k\in\mathbb{N}$ and $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \Gamma_1)\to \infty$.

$(e) \; \xi_k\in\Lambda_\omega$ for all $k\in\mathbb{N}$, $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \partial\Lambda_\omega)\to\infty$ and, either $\varepsilon_k^{-1}|\xi_k-\varrho\xi_k|\to\infty$, or $\xi_k = \varrho\xi_k$ for all $k\in\mathbb{N}$.

This can be seen as follows: If the sequence $(\varepsilon_k^{-1}|x_k|)$ is bounded, we set $\xi_k: = 0$. Then, (2.12) and $(a)$ hold true. If $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \partial\omega))$ is bounded, we take $\xi_k\in\partial\omega$ such that $|x_k-\xi_k| = \mathrm{dist}(x_k, \partial\omega)$. Then, (2.12) and $(b)$ hold true. If both $(\varepsilon_k^{-1}|x_k|)$ and $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \partial\omega))$ are unbounded and $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \Gamma_1))$ is bounded, we take $\xi_k\in\Gamma_1$ with $|x_k-\xi_k| = \mathrm{dist}(x_k, \Gamma_1)$. Then, (2.12) and $(c)$ hold true. If $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \Gamma_1))$ is unbounded and $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \omega))$ is bounded, we take $\xi_k\in\omega$ with $|x_k-\xi_k| = \mathrm{dist}(x_k, \omega)$. Then, (2.12) and $(d)$ hold true. Finally, if $(\varepsilon_k^{-1}\mathrm{dist}(x_k, \partial\Lambda_\omega))$ is unbounded, we set $\xi_k: = \frac{x_k+\varrho x_k}{2}$ if $(\varepsilon_k^{-1}|x_k-\varrho x_k|)$ is bounded and $\xi_k: = x_k$ if $(\varepsilon_k^{-1}|x_k-\varrho x_k|)$ is unbounded. Then, (2.12) and $(e)$ hold true.

Set $C_1: = C_0+1$. Inequality (2.12) yields

We consider $u_k$ as a function in $D^{1, 2}(\Sigma_\omega)$ via trivial extension, and we define $\widehat{u}_k\in D^{1, 2}(\Sigma_\omega)$ as $\widehat{u}_k(z): = \varepsilon_k^{(N-2)/2}u_k(\varepsilon_kz)$. Since $\widehat{u}_k$ is $\varrho$-equivariant, so is its extension $P_{\varepsilon_k}\widehat{u}_k\in D^{1, 2}(\mathbb{R}^N)$ given by Lemma 2.2. Let

Then,

and $(w_k)$ is bounded in $D^{1, 2}(\mathbb{R}^N)$. Hence, a subsequence satisfies that $w_k \rightharpoonup w$ weakly in $D^{1, 2}(\mathbb{R}^N)$, $w_k \to w$ a.e. in $\mathbb{R}^N$ and $w_k \to w$ strongly in $L^2_{\mathrm{loc}}(\mathbb{R}^N)$. Choosing $\delta$ sufficiently small and using (2.16), a standard argument shows that $w\neq 0$; see, e.g., [10,Section 8.3]. Moreover, we have that $\xi_k \to \xi$ and $\varepsilon_k \to 0$, because $u_k \rightharpoonup 0$ weakly in $V(\Lambda_{\omega})$ and $w\neq 0$.

Let $\mathbb{E}$ be the limit of the domains $\Lambda_k$. Since $(w_k)$ is bounded in $D^{1, 2}(\mathbb{R}^N)$, using Hölder's inequality we obtain

for every $\varphi\in\mathcal{C}^\infty_c(\mathbb{R}^N)$, and similarly for the integrals over $\Lambda_k\backslash \mathbb{E}$. Therefore, as $w_k \rightharpoonup w$ weakly in $D^{1, 2}(\mathbb{E})$, rescaling and using (2.14) we conclude that

where $\varphi_k(x): = \varepsilon_k^{(2-N)/2}\varphi(\frac{x-\xi_k}{\varepsilon_k})$. Next, we analyze all possibilities, according to the location of $\xi_k$.

$(a)$ If $\xi_k = 0$ for all $k\in\mathbb{N}$, then $\mathbb{E} = \Sigma_\omega$ and $w_k$ is $\varrho$-equivariant. Hence, $w$ is $\varrho$-equivariant. Let $\varphi\in\mathcal{C}^\infty_c(\mathbb{R}^N)$. Then, $\varphi_k|_{\Lambda_\omega}\in V(\Lambda_\omega)$ for large enough $k$, and from (2.17) we obtain

This shows that $w|_{\Sigma_\omega}$ solves (1.1). Therefore,

Together with Lemma 2.1, this implies that $J_{\Sigma_\omega}(w) = c^\varrho_{\Sigma_\omega} = c^\varrho_{\Lambda_\omega}\leq c_\infty$ and

So, in this case, we obtain statement $(ii)$.

$(b)$ If $\xi_k\in\partial\omega$ for all $k\in\mathbb{N}$, then $\mathbb{E} = \mathbb{H}_\xi\cap\mathbb{H}_\nu$, where $\xi = \lim_{k\to\infty}\xi_k$, \; $\nu$ is the exterior unit normal to $\Sigma_\omega$ at $\xi$, and $\mathbb{H}_\xi$ and $\mathbb{H}_\nu$ are half-spaces defined as in (2.7). If $\varphi\in\mathcal{C}^\infty_c(\mathbb{H}_\xi)$, then $\varphi_k|_{\Lambda_\omega}\in V(\Lambda_\omega)$ for large enough $k$, and using (2.17) we conclude that $w|_\mathbb{E}$ solves the mixed boundary value problem

Since $\xi$ and $\nu$ are orthogonal, extending $w|_\mathbb{E}$ by reflection on $\partial\mathbb{E}\cap\partial\mathbb{H}_\nu$, yields a nontrivial solution to the Dirichlet problem

It is well known that this problem does not have a nontrivial solution, so $(b)$ cannot occur.

$(c)$ If $\xi_k\in\Gamma_1$ for all $k\in\mathbb{N}$ and $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \bar{\omega}\cup\{0\})\to \infty$, then $\mathbb{E} = \mathbb{H}_\nu$, where $\nu$ is the exterior unit normal to $\Sigma_\omega$ at $\xi = \lim_{k\to\infty}\xi_k$. Using (2.17) we conclude that $w|_{\mathbb{H}_\nu}$ solves the Neumann problem (2.10) in $\mathbb{H}_\nu$. Since $\varepsilon_k^{-1}|\xi_k|\to\infty$, we have that $\varepsilon_k^{-1}|\xi_k-\varrho\xi_k|\to\infty$. Therefore,

Note also that $w_k\circ\varrho\rightharpoonup w\circ\varrho$ weakly in $D^{1, 2}(\mathbb{R}^{N})$. Using these facts and performing suitable rescalings and translations we obtain

Since $J_{\mathbb{H}_\nu}(w) = \frac{1}{N}\|w\|^2_{\mathbb{H}_\nu}\geq \frac{1}{2}c_\infty$, applying Lemma 2.1 we conclude that $J_{\mathbb{H}_\nu}(w) = \frac{1}{2}c_\infty$, $c_{\Sigma_\omega}^\varrho = c_{\Lambda_\omega}^\varrho = c_\infty$, and

So, in this case we obtain statement $(i)$.

$(d)$ If $\xi_k\in\omega$ for all $k\in\mathbb{N}$ and $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \Gamma_1)\to \infty$, then $\mathbb{E} = \mathbb{H}_\xi$ and using (2.17) we conclude that $w|_{\mathbb{H}_\xi}$ solves the Dirichlet problem (2.18). So this case does not occur.

$(e)$ If $\xi_k\in\Lambda_\omega$ for all $k\in\mathbb{N}$ and $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \partial\Lambda_\omega)\to\infty$, then $\mathbb{E} = \mathbb{R}^N$ and $w$ solves the problem (1.2). If $\rho\xi_k = \xi_k$ for every $k$, then $w_k$ is $\varrho$-equivariant, and so is $w$. Since $w$ is a sign-changing solution to (1.2) we have that

contradicting Lemma 2.1. On the other hand, if $\varepsilon_k^{-1}|\varrho\xi_k-\xi_k|\to\infty$, then, arguing as in case $(c)$, we conclude that

contradicting Lemma 2.1 again. So $(e)$ cannot occur.

We are left with $(a)$ and $(c)$. This concludes the proof.

Proposition 2.3 immediately yields the following result.

Corollary 2.4. If $c^{\varrho}_{\Sigma_\omega} < c_\infty$, then the problem (1.1) has a $\varrho$-equivariant least energy solution in $D^{1, 2}(\Sigma_\omega)$.

Equality is not enough, as the following example shows. Set

Example 2.5. If $\omega = \mathbb{S}^{N-1}_+$, then problem (1.1) does not have a $\varrho$-equivariant least energy solution in $D^{1, 2}(\Sigma_\omega)$.

Proof. $\Sigma_\omega$ is the upper half-space $\mathbb{R}^N_+: = \{(x_1, \ldots, x_N)\in\mathbb{R}^N:x_N > 0\}$. If $u$ were a $\varrho$-equivariant least energy solution to (1.1) in $\mathbb{R}^N_+$ then, extending $u$ by reflection on $\partial(\mathbb{R}^N_+)$, would yield a sign-changing solution $\widetilde{u}$ to the problem (1.2) in $\mathbb{R}^N$ with $J_{\mathbb{R}^N}(\widetilde{u})\leq 2c_\infty$. But the energy of any sign-changing solution to (1.2) is $> 2c_\infty$; see ^[9].

The following local geometric condition guarantees the existence of a minimizer. It was introduced by Adimurthi and Mancini in ^[1].

Definition 2.6. A point $\xi\in\partial\omega$ is a point of convexity of $\Sigma_\omega$ (of radius $r > 0$) if $B_r(\xi)\cap\Sigma_\omega\subset\mathbb{H}_\nu$ and the mean curvature of $\partial\Sigma_\omega$ at $\xi$ with respect to the exterior unit normal $\nu$ at $\xi$ is positive.

As in ^[1] we make the convention that the curvature of a geodesic in $\partial\Sigma_\omega$ is positive at $\xi$ if it curves away from the exterior unit normal $\nu$. The half-space $\mathbb{H}_\nu$ is defined as in (2.7). Examples of cones having a point of convexity are given as follows.

Proposition 2.7. If $\overline{\omega}\subset\mathbb{S}^{N-1}_+$, then $\Sigma_\omega$ has a point of convexity.

Proof. Let $\beta$ be the smallest geodesic ball in $\mathbb{S}^{N-1}$, centered at the north pole $(0, \ldots, 0, 1)$, which contains $\omega$. Then, $\partial\omega\cap\partial\beta\neq\emptyset$ and $\overline{\beta}\subset\mathbb{S}^{N-1}_+$. Hence, every point on $\partial\beta$ is a point of convexity of $\Sigma_\beta$. As $\omega\subset\beta$, we have that any point $\xi\in\partial\omega\cap\partial\beta$ is a point of convexity of $\Sigma_\omega$.

Theorem 2.8. If $\Sigma_\omega$ has a point of convexity, then $c^{\varrho}_{\Sigma_\omega} < c_\infty$. Consequently, the problem (1.1) has a $\varrho$-equivariant least energy solution in $D^{1, 2}(\Sigma_\omega)$. This solution is nonradial and changes sign.

Proof. Let $\xi\in\partial\omega$ be a point of convexity of $\Sigma_\omega$ of radius $r > 0$. It is shown in [1,Lemma 2.2] that, after fixing $r$ small enough and a radial cut-off function $\psi\in\mathcal{C}^\infty_c(\mathbb{R}^N)$ with $\psi(x) = 1$ if $|x|\leq \frac{r}{4}$ and $\psi(x) = 0$ if $|x|\geq \frac{r}{2}$, the function $u_{\varepsilon, \xi}(x): = \psi(x-\xi)\varepsilon^{(2-N)/2}U(\varepsilon^{-1}(x-\xi))$, with $U$ as in (2.3), satisfies

where $d_N$ is a positive constant depending only on $N$ and $H_\omega(\xi)$ is the mean curvature of $\partial\Sigma_\omega$ at $\xi$. Hence, for $\varepsilon$ small enough,

where $t_{\varepsilon, \xi} > 0$ is such that $t_{\varepsilon, \xi}u_{\varepsilon, \xi}\in\mathcal{N}(\Sigma_\omega)$; see (2.1). Choosing $r$ so that $B_r(\xi)\cap B_r(\varrho\xi) = \emptyset$ we conclude that $t_{\varepsilon, \xi}(u_{\varepsilon, \xi}-u_{\varepsilon, \xi}\circ\varrho)\in\mathcal{N}^\varrho(\Sigma_\omega)$ and

The existence of a $\varrho$-equivariant least energy solution to (1.1) follows from Corollary 2.4.

3.   A positive nonradial solution

In this section $\omega$ is not assumed to have any symmetries.

We are interested in positive solutions to the problem (1.1). Note that this problem has always a positive radial solution given by the restriction to $\Sigma_\omega$ of the standard bubble $U$ defined in (2.3). The question we wish to address in this section is whether problem (1.1) has a positive nonradial solution.

Recall the notation introduced in Section 2 and set

It is shown in [8,Theorem 2.1] that $c_{\Lambda_\omega} > 0$. As in Lemma 2.1 one shows that $c_{\Sigma_\omega} = c_{\Lambda_\omega}\leq\frac{1}{2} c_\infty$. We start by describing the behavior of minimizing sequences for $J_{\Lambda_\omega}$ on $\mathcal{N}(\Lambda_\omega)$.

Proposition 3.1. Let $u_k\in\mathcal{N}(\Lambda_\omega)$ be such that

Then, after passing to a subsequence, one of the following statements holds true:

$(i)$ There exist a sequence of positive numbers $(\varepsilon_k)$, a sequence of points $(\xi_k)$ in $\Gamma_1$ and a function $w\in D^{1, 2}(\mathbb{R}^N)$ such that $\varepsilon_k^{-1}\mathrm{dist}(\xi_k, \bar{\omega}\cup\{0\})\to\infty$, $w|_\mathbb{H}$ solves the Neumann problem

in some half-space $\mathbb{H}$, $J_{\mathbb{H}}(w) = \frac{1}{2}c_\infty$,

and $c_{\Sigma_\omega} = c_{\Lambda_\omega} = \frac{1}{2} c_\infty$.

$(ii)$ There exist a sequence of positive numbers $(\varepsilon_k)$ with $\varepsilon_k\to 0$ and a solution $w\in D^{1, 2}(\Sigma_\omega)$ to the problem (1.1) such that

and $J_{\Sigma_\omega}(w) = c_{\Sigma_\omega} = c_{\Lambda_\omega}\leq\frac{1}{2} c_\infty$.

Proof. The proof is similar, but simpler than that of Proposition 2.3.

The following statement is an immediate consequence of this proposition.

Corollary 3.2. If $c_{\Sigma_\omega} < \frac{1}{2}c_\infty$, then the problem (1.1) has a positive least energy solution in $D^{1, 2}(\Sigma_\omega)$.

Theorem 3.3. If $\Sigma_\omega$ has a point of convexity, then $c_{\Sigma_\omega} < \frac{1}{2} c_\infty$. Consequently, the problem (1.1) has a positive least energy solution in $D^{1, 2}(\Sigma_\omega)$.

Proof. The proof is similar to that of Theorem 2.8.

Let $D^{1, 2}_\mathrm{rad}(\Sigma_\omega)$ be the subspace of radial functions in $D^{1, 2}(\Sigma_\omega)$, and define $\mathcal{N}^\mathrm{rad}(\Sigma_\omega): = \mathcal{N}(\Sigma_\omega)\cap D^{1, 2}_\mathrm{rad}(\Sigma_\omega)$ and

It was shown in [8,Theorem 2.4] that, if $\Sigma_\omega$ is convex, then $c_{\Sigma_\omega}^\mathrm{rad} = c_{\Sigma_\omega}$ and the only positive minimizers are the restrictions of the rescalings

of the standard bubble to $\Sigma_\omega$. In fact, the proof of [8,Theorem 2.4] shows that these are the only positive solutions of (1.1) in a convex cone. Moreover, the following statement holds true.

Proposition 3.4. For any cone $\Sigma_\omega$, the restrictions to $\Sigma_\omega$ of the functions $U_\varepsilon$ defined in (3.1) are minimizers of $J_{\Sigma_\omega}$ on $\mathcal{N}^\mathrm{rad}(\Sigma_\omega)$. These are the only nontrivial radial solutions to (1.1), up to sign. Moreover,

and $|X|$ is the Lebesgue measure of $X$. In particular, $c_{\Sigma_\omega}^\mathrm{rad}$ increases with $|\Lambda_\omega|$.

Proof. A radial function $u$ solves (1.1) in $\Sigma_\omega$ if and only if the function $\bar{u}$ given by $\bar{u}(r): = u(x)$ with $r = \|x\|$ solves

This last problem does not depend on $\omega$. It is well known that, up to sign, the functions $U_\varepsilon$ are the only nontrivial radial solutions to the problem (1.2) in $\mathbb{R}^N = \Sigma_{\mathbb{S}^{N-1}}$. Hence, their restrictions to $\Sigma_\omega$ are the only nontrivial radial solutions to (1.1).

As in Lemma 2.1 one shows that $c_{\Sigma_\omega}^\mathrm{rad} = c_{\Lambda_\omega}^\mathrm{rad}: = \inf_{u\in\mathcal{N}^\mathrm{rad}(\Lambda_\omega)}J_{\Lambda_\omega}(u)$. For $u\in V_\mathrm{rad}(\Lambda_\omega): = D^{1, 2}_\mathrm{rad}(\Lambda_\omega)\cap V(\Lambda_\omega), u\neq 0$, we have that

Therefore,

The same formula holds true when we replace $\omega$ by $\mathbb{S}^{N-1}$. In this case, the left-hand side is $c_\infty$. Hence, $b_N = \frac{c_\infty}{|B_1(0)|}$, as claimed.

Corollary 3.5. If $\Sigma_\omega$ has a point of convexity and $|\Lambda_\omega|\geq\frac{1}{2}|B_1(0)|$, then

$(i)$ the problem (1.1) has a positive least energy solution in $D^{1, 2}(\Sigma_\omega)$,

$(ii)$ every least energy solution of (1.1) is nonradial.

Proof. From Theorem 3.3 and Proposition 3.4 we get that $c_{\Sigma_\omega}$ is attained and

where $\mathbb{R}^N_+: = \{(x_1, \ldots, x_N)\in\mathbb{R}^N:x_N > 0\}$. So every least energy solution is nonradial.

Note that the hypothesis that $|\Lambda_\omega|\geq\frac{1}{2}|B_1(0)|$ implies that $\Sigma_\omega$ is not convex.

A closer look at the estimate (2.19) allows to refine Corollary 3.5 and to produce examples of cones $\Sigma_\omega$ with $|\Lambda_\omega| < \frac{1}{2}|B_1(0)|$ for which the problem (1.1) has a positive nonradial solution.

To this end, we fix a smooth domain $\omega_0$ in $\mathbb{S}^{N-1}$ for which $\Sigma_{\omega_0}$ has a point of convexity $\xi\in\partial\omega_0$ of radius $r > 0$, and we define

Then, we have the following result.

Theorem 3.6. There exists $\alpha_\xi\in (0, \frac{1}{2}|B_1(0)|)$, depending only on $B_r(\xi)\cap\Sigma_{\omega_0}$, such that, for every $\omega\in \ell(\omega_0, \xi, r)$ with $|\Lambda_\omega| > \alpha_\xi$, the following statements hold true:

$(i)$ the problem (1.1) has a positive least energy solution in $D^{1, 2}(\Sigma_\omega)$,

$(ii)$ every least energy solution of (1.1) is nonradial,

$(iii)\; \Sigma_\omega$ is not convex.

Proof. Recall that the functions $u_{\varepsilon, \xi}$, introduced in the proof of Theorem 2.8, vanish outside the ball $B_{r/2}(0)$. Moreover, the value $Q_{\Sigma_{\omega_0}}(u_{\varepsilon, \xi})$ and the estimate (2.19) depend only on the value of $u_{\varepsilon, \xi}$ in $B_r(\xi)\cap\Sigma_{\omega_0 }$. We fix $\varepsilon_0 > 0$ small enough so that

and we set $\alpha_\xi: = \frac{1}{Nb_N}Q_\xi^{N/2}$ with $b_N$ as in Proposition 3.4. Then,

Given $\omega\in \ell(\omega_0, \xi, r)$, we fix a function $\widehat{u}_{\varepsilon_0, \xi}\in\mathcal{C}_c^\infty(B_r(0))$ such that $\widehat{u}_{\varepsilon_0, \xi}(x) = u_{\varepsilon_0, \xi}(x)$ if $x\in B_r(\xi)\cap\Sigma_{\omega_0}$ and $\widehat{u}_{\varepsilon_0, \xi}(x) = 0$ if $x\in B_r(\xi)\cap(\Sigma_{\omega}\backslash \Sigma_{\omega_0})$. So, if $|\Lambda_\omega| > \alpha_\xi$, we have that

Note that $\xi$ is a point of convexity of $\omega$. Hence, by Theorem 3.3 and the previous inequality, $c_{\Sigma_\omega}$ is attained at a nonradial solution of (1.1). Finally, recall that, if $\Sigma_\omega$ were convex, then $c_{\Sigma_\omega} = c_{\Sigma_\omega}^\mathrm{rad}$; see [8,Theorem 2.4]. This completes the proof.

Corollary 3.7. There exists a smooth domain $\omega\subset\mathbb{S}^{N-1}_+$ such that the problem (1.1) has a positive nonradial solution in $\Sigma_\omega$.

Proof. Let $\omega_0$ be the geodesic ball in $\mathbb{S}^{N-1}$ of radius $\pi/4$ centered at the north pole and let $\xi$ be any point on $\partial\omega_0$. Fix $r > 0$ such that $B_r(\xi)\cap\mathbb{S}^{N-1}\subset\mathbb{S}^{N-1}_+$. Clearly, $\xi$ is a point of convexity of $\Sigma_{\omega_0}$ of radius $r$, so we may fix $\alpha_\xi > 0$ as in Theorem 3.6. As $\alpha_\xi < \frac{1}{2}|B_1(0)|$, there exists $\omega\in \ell(\omega_0, \xi, r)$ with $\omega\subset\mathbb{S}^{N-1}_+$ and $|\Lambda_\omega| > \alpha_\xi$. Now, Theorem 3.6 yields a positive nonradial solution to problem (1.1) in $\Sigma_\omega$.

Remark 3.8. Let $\omega\neq\mathbb{S}^{N-1}_+$ be such that $\Sigma_\omega$ is convex. Then, every point $\xi\in\partial\omega$ is a point of convexity of radius $r$ for any $r > 0$. Fix $r = 1$, and fix $\varepsilon > 0$ such that

Now, define $\alpha_\xi: = \frac{1}{Nb_N}Q_\xi^{N/2}$, as in Theorem 3.6. Since $\Sigma_\omega$ is convex, we must have that

where the equality follows from the definition of $b_N$; see Proposition 3.4. Hence, for any convex cone $\Sigma_\omega$, we obtain the upper bound

for the measure of $\Lambda_\omega$, which is given in terms of the Sobolev constant and the local energy of the standard bubbles.

Acknowledgments

M. Clapp was partially supported by UNAM-DGAPA-PAPIIT grant IN100718 (Mexico) and CONACYT grant A1-S-10457 (Mexico). F. Pacella was partially supported by PRIN 2015 (Italy) and INDAM-GNAMPA (Italy).

Conflict of interest

The authors declare no conflict of interest.