Multiple solutions for superlinear fractional problems via theorems of mixed type

Abstract

In this paper we investigate the existence of multiple solutions for the following two fractional problems \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\Omega})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \partial \Omega \end{array} \right. \end{equation*} and \begin{equation*} \left\{\begin{array}{ll} (-\Delta_{\mathbb{R}^{N}})^{s} u-\lambda u= f(x, u) &\mbox{in} \Omega \\ u=0 &\mbox{in} \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N>2s$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$, and $f:\bar{\Omega}\times \mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous function which does not satisfy the well-known Ambrosetti-Rabinowitz condition. Here $(-\Delta_{\Omega})^{s}$ is the spectral Laplacian and $(-\Delta_{\mathbb{R}^{N}})^{s}$ is the fractional Laplacian in $\mathbb{R}^{N}$. By applying variational theorems of mixed type due to Marino and Saccon and Linking Theorem, we prove the existence of multiple solutions for the above problems.