Multiplicity Results of a Nonlocal Problem Involving Concave-Convex Nonlinearities

Abstract

In this work, we investigate the following fractional $p$-Laplacian equation involving a concave-convex nonlinearities as follows, ${\rm (P_\lambda)} \begin{cases} (-\Delta)_p^s u = \lambda u^{q} + u^{r} &\mbox{in }\Omega, \\ u>0 & \text{in }\Omega, \\ u = 0 &\mbox{in }\mathbb{R}^N\setminus\Omega, \end{cases}$ where $\Omega\subset\mathbb{R}^N$, $N\geq 2$ is a bounded domain with $C^{1,1}$ boundary $\partial\Omega,$ $\lambda >0$, $1<p<\infty,$ $s\in (0,1)$ such that $N\geq s p,$ $0<q<p-1<r\leq p^*_s-1,$ $p^*_s = \frac{Np}{N-s p}$ is the fractional critical Sobolev exponent and the nonlinear nonlocal operator $(-\Delta)^s_p u$ with $s\in (0,1)$ is the $p$-fractional Laplacian defined on smooth functions by $(-\Delta)^s_p u(x)=2 \underset{\epsilon\searrow 0}{\lim}\int_{\mathbb{R}^{N}\backslash B_\epsilon}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ ps}}\, dy,\qquad x\in \mathbb{R}^N.$ We use variational methods, in order to show the existence of multiple positive solutions to the problem $(P_\lambda)$ for different value of $\lambda.$