Multiplicity results for Schr\"odinger 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} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in \R^{N}\backslash\Omega, \end{aligned} \right. \end{equation} where $\Omega\subset\R^{N}$ is an open bounded set with Lipschitz boundary $\partial\Omega$, $N\geqslant 2,$ $V\in L^{\infty}(\R^{N})$, and $(-\Delta)_p^s$ denotes the fractional $p$-Laplacian with $s\in(0,1), 1<p$, $sp<N$, $\lambda>0$, and $f:\R\rightarrow\R$ 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.