Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

Abstract

We consider the following nonlocal critical problem with mixed Dirichlet-Neumann boundary conditions, \begin{equation} \left\{ \begin{array}{ll} (-Δ)^su=λu+|u|^{2_s^*-2}u &\text{in}\ Ω,\\ \mkern+38.5mu u=0& \text{on}\ Σ_{\mathcal{D}},\\ \mkern+24mu \displaystyle \frac{\partial u}{\partial ν}=0 &\text{on}\ Σ_{\mathcal{N}}, \end{array} \right. \end{equation} where $(-\Delta {)}^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $\Omega \subset {\mathbb{R}}^N$, $N>2s$, is a smooth bounded domain, $2_s^{\ast }=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $\lambda >0$ is a real parameter, $\nu$ is the outwards normal to $\partial \Omega$, ${\Sigma }_{\mathcal{D}}$, ${\Sigma }_{\mathcal{N}}$ are smooth $(N-1)$--dimensional submanifolds of $\partial \Omega$ such that ${\Sigma }_{\mathcal{D}}\cup {\Sigma }_{\mathcal{N}}=\partial \Omega$, ${\Sigma }_{\mathcal{D}}\cap {\Sigma }_{\mathcal{N}}=\varnothing$ and ${\Sigma }_{\mathcal{D}}\cap {\overline{\Sigma }}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$--dimensional submanifold of $\partial \Omega$. By employing a $\nabla$-theorem we prove the existence of multiple solutions when the parameter $\lambda$ is in a left neighborhood of a given eigenvalue of $(-\Delta {)}^s$.

18 pages