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} (-Δ_Ω)^{s} u-λu= f(x, u) &\mbox{in} Ω\\ u=0 &\mbox{in} \partial Ω\end{array} \right. \end{equation*} and \begin{equation*} \left\{\begin{array}{ll} (-Δ_{\mathbb{R}^{N}})^{s} u-λu= f(x, u) &\mbox{in} Ω\\ u=0 &\mbox{in} \mathbb{R}^{N}\setminus Ω, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N>2s$, $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^N$, and $f:\bar{\Omega }\times \mathbb{R}\to \mathbb{R}$ 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.

Adv. Nonlinear Stud. (2018)

References in corpus (2)

• Existence and multiplicity results for the fractional Laplacian in bounded domains
• Periodic solutions for critical fractional problems