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

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u&\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0&\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} 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 $2^*:=\frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $\omega$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $\Sigma_\omega\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.


Introduction
We consider the Neumann problem For u ∈ D 1,2 (Ω) {0} let t u ∈ (0, ∞) be such that t u u ∈ N (Ω). Then, where Q Ω (u) := Ω |∇u| 2 Ω |u| 2 * 2/2 * . (2.1) Hence, J Ω (u) = inf We set c ∞ := c R N . It is well known that this infimum is attained at the function U (x) = a N 1 1 + |x| 2 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 (R N ) ֒→ L 2 * (R N ). Let S N −1 be the unit sphere in R N and let ω be a smooth domain in S N −1 with nonempty boundary, i.e., ω is connected and open in S N −1 and its boundary ∂ω is a smooth (N − 2)-dimensional submanifold of S N −1 . The nontrivial solutions to the Neumann problem (1.1) in the unbounded cone Σ ω := {tx : x ∈ ω and t > 0} are the critical points of J Σω on N (Σ ω ).
To produce a nonradial sign-changing solution for (1.1) we introduce some symmetries. We write a point in R N as x = (x ′ , x N ) ∈ R N −1 × R, and consider the reflection ̺(x ′ , x N ) := (−x ′ , x N ). Then, a subset X of R N will be called ̺-invariant if ̺x ∈ X for every x ∈ X, and a function u : X → R will be called ̺-equivariant if u(̺x) = −u(x) ∀x ∈ X.
Note that every nontrivial ̺-equivariant function is nonradial and changes sign.
Throughout this section we will assume that ω is ̺-invariant. Note that (0, ±1) ∈ ∂ω because ∂ω is smooth. Hence, ̺x = x for every x ∈ ∂Σ ω {0}. Our aim is to show that (1.1) has a ̺-equivariant solution. We set and set Γ 1 := ∂Λ ω ω. In Λ ω 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 As s · ν = 0 for every s ∈ Γ 1 and s · ν > 0 for every s ∈ ω, we conclude that ∂u ∂ν vanishes on ω. Therefore, the trivial extension of u to the infinite cone Σ ω solves (1.1), contradicting the unique continuation principle. Let V (Λ ω ) be the space of functions in D 1,2 (Λ ω ) whose trace vanishes on ω. Note that V (Λ ω ) ⊂ D 1,2 (Σ ω ) via trivial extension. Let J Λω : V (Λ ω ) → R be the restriction of J Σω to V (Λ ω ) and set and To produce a sign-changing solution for the problem (1.1) we will study the concentration behavior of ̺-equivariant minimizing sequences for (2.5). We start with the following lemmas.
To prove the opposite inequality, let ϕ k ∈ N ̺ (Σ ω )∩C ∞ (Σ ω ) be such that ϕ k has compact support and J(ϕ k ) → c ̺ Σω as k → ∞. Then, we may choose ε k > 0 such that the support of Letting k → ∞ we conclude that c ̺ Λω ≤ c ̺ Σω . To prove that c ̺ Σω ≤ c ∞ we fix a point ξ ∈ ∂Σ ω {0} and a sequence of positive numbers ε k → 0, and we set Σ k := ε −1 k (Σ ω − ξ). Since ∂Σ ω {0} is smooth, the limit of the sequence of sets (Σ k ) is the half-space where ν is the exterior unit normal to Σ ω at ξ. Let u k (x) := ε The function This concludes the proof. If ∂Ω is Lipschitz continuous, then there exist linear extension operators P ε : W 1,2 (Ω ε ) → D 1,2 (R N ) and a positive constant C, independent of ε, such that Proof. The existence of an extension operator P ε : is well known, and the fact that the constant C does not depend on ε was proved in [6, Lemma 2.1]. To obtain (iv) we replace P ε u by the function The following proposition describes the behavior of minimizing sequences for J Λω on N ̺ (Λ ω ).
Then, after passing to a subsequence, one of the following statements holds true: (i) There exist a sequence of positive numbers (ε k ), a sequence of points (ξ k ) in Γ 1 and a function w the sequence (u k ) is bounded and, after passing to a subsequence, u k ⇀ u weakly in V (Λ ω ). Then, J ′ Λω (u) = 0. Since the problem (2.5) does not have a nontrivial solution, we conclude that u = 0.
Fix δ ∈ (0, N 2 c ̺ Λω ). As there are bounded sequences (ε k ) in (0, ∞) and (x k ) in R N such that, after passing to a subsequence, We claim that, after passing to a subsequence, there exist ξ k ∈Λ ω and C 0 > 0 such that and one of the following statements holds true: (a) ξ k = 0 for all k ∈ N.
This can be seen as follows: If the sequence (ε −1 k |x k |) is bounded, we set ξ k := 0. Then, (2.12) and (a) hold true.
Then, (2.12) and (e) hold true. Set We consider u k as a function in D 1,2 (Σ ω ) via trivial extension, and we define and (w k ) is bounded in D 1,2 (R N ). Hence, a subsequence satisfies that w k ⇀ w weakly in D 1,2 (R N ), w k → w a.e. in R N and w k → w strongly in L 2 loc (R N ). Choosing δ sufficiently small and using (2.16), a standard argument shows that w = 0; see, e.g., [10,Section 8.3]. Moreover, we have that ξ k → ξ and ε k → 0, because u k ⇀ 0 weakly in V (Λ ω ) and w = 0.
Let E be the limit of the domains Λ k . Since (w k ) is bounded in D 1,2 (R N ), using Hölder's inequality we obtain for every ϕ ∈ C ∞ c (R N ), and similarly for the integrals over Λ k E. Therefore, as w k ⇀ w weakly in D 1,2 (E), rescaling and using (2.14) we conclude that Next, we analyze all possibilities, according to the location of ξ k .
(b) If ξ k ∈ ∂ω for all k ∈ N, then E = H ξ ∩ H ν , where ξ = lim k→∞ ξ k , ν is the exterior unit normal to Σ ω at ξ, and H ξ and H ν are half-spaces defined as in (2.7). If ϕ ∈ C ∞ c (H ξ ), then ϕ k | Λω ∈ V (Λ ω ) for large enough k, and using (2.17) we conclude that w| E solves the mixed boundary value problem Since ξ and ν are orthogonal, extending w| E by reflection on ∂E ∩ ∂H ν , 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 ξ k ∈ Γ 1 for all k ∈ N and ε −1 k dist(ξ k ,ω ∪ {0}) → ∞, then E = H ν , where ν is the exterior unit normal to Σ ω at ξ = lim k→∞ ξ k . Using (2.17) we conclude that w| Hν solves the Neumann problem (2.10) in H ν . Since Note also that w k • ̺ ⇀ w • ̺ weakly in D 1,2 (R N ). Using these facts and performing suitable rescalings and translations we obtain So, in this case we obtain statement (i).
(e) If ξ k ∈ Λ ω for all k ∈ N and ε −1 k dist(ξ k , ∂Λ ω ) → ∞, then E = R N and w solves the problem (1.2). If ρξ k = ξ k for every k, then w k is ̺-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 ε −1 k |̺ξ k − ξ k | → ∞, 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.
Equality is not enough, as the following example shows. Set But the energy of any sign-changing solution to (1.2) is > 2c ∞ ; 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 ξ ∈ ∂ω is a point of convexity of Σ ω of radius r > 0 if B r (ξ) ∩ Σ ω ⊂ H ν and the mean curvature of ∂Σ ω at ξ with respect to the exterior unit normal ν at ξ is positive.
As in [1] we make the convention that the curvature of a geodesic in ∂Σ ω is positive at ξ if it curves away from the exterior unit normal ν. The half-space H ν is defined as in (2.7). Examples of cones having a point of convexity are given as follows. Proof. Let β be the smallest geodesic ball in S N −1 , centered at the north pole (0, . . . , 0, 1), which contains ω. Then, ∂ω ∩ ∂β = ∅ and β ⊂ S N −1 + . Hence, every point on ∂β is a point of convexity of Σ β . As ω ⊂ β, we have that any point ξ ∈ ∂ω ∩ ∂β is a point of convexity of Σ ω . Theorem 2.8. If Σ ω has a point of convexity, then c ̺ Σω < c ∞ . Consequently, the problem (1.1) has a ̺-equivariant least energy solution in D 1,2 (Σ ω ). This solution is nonradial and changes sign.

A positive nonradial solution
In this section ω 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 Σ ω 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 Λω > 0. As in Lemma 2.1 one shows that c Σω = c Λω ≤ 1 2 c ∞ . We start by describing the behavior of minimizing sequences for J Λω on N (Λ ω ).
Then, after passing to a subsequence, one of the following statements holds true: (i) There exist a sequence of positive numbers (ε k ), a sequence of points (ξ k ) in Γ 1 and a function w and c Σω = c Λω = 1 2 c ∞ .
(ii) There exist a sequence of positive numbers (ε k ) with ε k → 0 and a solution w ∈ D 1,2 (Σ ω ) to the problem (1.1) such that The proof is similar, but simpler than that of Proposition 2.3.
The following statement is an immediate consequence of this proposition.  Proof. The proof is similar to that of Theorem 2.8.
and |X| is the Lebesgue measure of X. In particular, c rad Σω increases with |Λ ω |. This last problem does not depend on ω. It is well known that, up to sign, the functions U ε are the only nontrivial radial solutions to the problem (1.2) in R N = Σ S N −1 . Hence, their restrictions to Σ ω are the only nontrivial radial solutions to (1.1). As in Lemma 2.1 one shows that c rad Σω = c rad Λω := inf u∈N rad (Λω ) J Λω (u). For Therefore, The same formula holds true when we replace ω by S N −1 . In this case, the left-hand side is c ∞ . Hence, b N = c∞ |B1(0)| , as claimed.
To this end, we fix a smooth domain ω 0 in S N −1 for which Σ ω0 has a point of convexity ξ ∈ ∂ω 0 of radius r > 0, and we define Then, we have the following result.
Now, define α ξ := 1 N bN Q N/2 ξ , as in Theorem 3.6. Since Σ ω 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 Σ ω , we obtain the upper bound for the measure of Λ ω , which is given in terms of the Sobolev constant and the local energy of the standard bubbles.