Multiplicity results for Schrödinger type fractional $p$-Laplacian boundary value problems

Abstract

In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(Δ)_{p}^{s} u + V(x)|u|^{p-2}u &= λf(u),&x\inΩ; u&=0,&x\in \R^{N}\backslashΩ, \end{aligned} \right. \end{equation} where $\Omega \subset {\mathbb{R}}^N$ is an open bounded set with Lipschitz boundary $\partial \Omega$, $N⩾2,$ $V\in L^{\infty }({\mathbb{R}}^N)$, and $(-\Delta {)}_p^s$ denotes the fractional $p$-Laplacian with $s\in (0,1),1<p$, $sp<N$, $\lambda >0$, and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function. We extend the results of Lopera {\it et al.} in \cite{Lopera1} by proving the existence of a second weak solution for problem (\ref{probaa1}). We apply a variant of the mountain-pass theorem due to Hofer \cite{Hofer2} and infinite-dimensional Morse theory to obtain the existence of at least two solutions.