Multiplicity of Nontrivial Solutions of a Class of Fractional $p$-Laplacian Problem

Abstract

In this paper, we deal with existence of nontrivial solutions to the fractional $p$-Laplacian problem of the type

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\partial \Omega$, $a\in C(\Omega )$, $p\ge 2$, $\alpha \in (0,1)$ such that $p\alpha <n$, $1<q<p<r<\frac{np}{n-\alpha p}$, and $F\in C^1(\overline{\Omega }\times \mathbb{R},\mathbb{R})$. Using the decomposition of the Nehari manifold, we prove that the non-local elliptic problem has at least two nontrivial solutions.