Subcritical nonlocal problems with mixed boundary conditions

Abstract

In this paper, by variational and topological arguments based on linking and $\nabla$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$\{\begin{array}{lcl}(-\Delta {)}^{s}u=\lambda u+f(x,u)& & \text{in }\Omega ,\\ u=0& & \text{on }{\Sigma }_{\mathcal{D}},\\ \frac{\partial u}{\partial \nu }=0& & \text{on }{\Sigma }_{\mathcal{N}},\end{array}$$ 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, $\lambda >0$ is a real parameter, $\nu$ is the outward 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 $\partial \Omega$.