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

TL;DR

The paper addresses multiplicity for a boundary value problem driven by the fractional $p$-Laplacian with a potential on a bounded Lipschitz domain. Using a Hofer-type mountain-pass theorem and infinite-dimensional Morse theory, the authors establish at least two weak solutions for small $\lambda$, and under extra assumptions obtain a positive solution as well. They further compute the critical groups at infinity and at the origin to justify the existence of multiple critical points, extending prior results for the fractional and nonlocal setting. The results rely on a variational framework, PS condition, and regularity/comparison principles to control sign and positivity of solutions, with the Appendix proving a key positivity result for an auxiliary problem.

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 $Ω\subset\R^{N}$ is an open bounded set with Lipschitz boundary $\partialΩ$, $N\geqslant 2,$ $V\in L^{\infty}(\R^{N})$, and $(-Δ)_p^s$ denotes the fractional $p$-Laplacian with $s\in(0,1), 1<p$, $sp<N$, $λ>0$, 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.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6