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

Abstract

We study the critical Neumann problem \begin{equation*} \begin{cases} -Δu = |u|^{2^*-2}u &\text{in }Σ_ω,\\ \quad\frac{\partial u}{\partialν}=0 &\text{on }\partialΣ_ω, \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, $N\ge 3$ and $2^{\ast }:=\frac{2N}{N-2}$. 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.