Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function

Abstract

In this article, we study the following $p$-fractional Laplacian equation \begin{equation*} (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dy = \la |u(x)|^{p-2}u(x) + b(x)|u(x)|^{\ba-2}u(x)\; \text{in}\; \Om \quad \quad\quad\quad \quad\quad\quad\quad\quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om,\quad u\in W^{\al,p}(\mb R^n).\\ \end{array} \quad \right. \end{equation*} where $\Om$ is a bounded domain in $\mb R^n$ with smooth boundary, $n> p\al$, $p\geq 2$, $\al\in(0,1)$, $\la>0$ and $b:\Om\subset\mb R^n \ra \mb R$ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of $(P_{\la})$ with respect to the parameter $\la$, which changes according to whether $1<\ba<p$ or $p< \ba< p^{*}=\frac{np}{n-p\al}$ respectively. We discuss both the cases separately. Non-existence results are also obtained.