Multiple solutions for superlinear fractional problems via theorems of mixed type

TL;DR

This paper establishes the existence of multiple solutions for superlinear fractional Laplacian problems in bounded and unbounded domains using advanced variational theorems, without relying on the classical Ambrosetti-Rabinowitz condition.

Contribution

It introduces new multiplicity results for fractional problems via theorems of mixed type, extending variational methods to cases lacking the Ambrosetti-Rabinowitz condition.

Findings

Multiple solutions are proven to exist for the fractional problems.

The results apply to both bounded and unbounded domains.

Variational methods of mixed type are effectively utilized.

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.