Existence of nonnegative solutions for a class of fractional p-Laplacian problems

Abstract

In this paper, we establish the existence of nonnegative solution to the following problem $$M\left(\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+ps}}dxdy\right)(-\triangle)_{p}^{s}u=f(x,u),\;\;\mbox{in}\;\;\Omega,$$
where $n>sp$ with $s\in (0,1)$, $\Omega$ is an open bounded subset of $\mathbb{R}^{n}$ with Lipschitz boundary, M and f are two continuous functions, and $(-\triangle)_{p}^{s}$ is a fractional p-laplacian operator. our main tools are based on critical point theorems and the truncation technique.