Existence and Multiplicity of Solutions for Fractional $p$-Laplacian Equation Involving Critical Concave-convex Nonlinearities

TL;DR

This paper studies the existence and multiplicity of solutions for a fractional p-Laplacian equation with critical and subcritical nonlinearities, revealing conditions for positive solutions and infinitely many solutions as parameters vary.

Contribution

It establishes a dichotomy for positive solutions based on the parameter lambda and proves the existence of multiple solutions for small lambda without sign restrictions.

Findings

Existence of positive solutions depending on lambda

Two positive solutions for small lambda when p≥2 and q in (p-1,p)

Infinitely many solutions exist for small lambda without sign constraints

Abstract

We investigate the following fractional $p$-Laplacian equation \[ \begin{cases} \begin{aligned} (-\Delta)_p^s u&=\lambda |u|^{q-2}u+|u|^{p_s^*-2}u &&\text{in}~\Omega,\\ u &=0 &&\text{in}~ \mathbb{R}^n\setminus\Omega, \end{aligned} \end{cases} \] where $s\in (0,1)$, $p>q>1$, $n>sp$, $\lambda>0$, $p_s^*=\frac{np}{n-sp}$ and $\Omega$ is a bounded domain (with $C^{1, 1}$ boundary). Firstly, we get a dichotomy result for the existence of positive solution with respect to $\lambda$. For $p\ge 2$, $p-1<q<p$, $n>\frac{sp(q+1)}{q+1-p}$, we provide two positive solutions for small $\lambda$. Finally, without sign constraint, for $\lambda$ sufficiently small, we show the existence of infinitely many solutions.