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In this article, we study the following $p$-fractional Laplacian equation  \begin{equation*}  (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mbR^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 smoothboundary, $n> p\al$, $p\geq 2$, $\al\in(0,1)$, $\la>0$ and $b:\Om\subset\mb R^n\ra \mb R$ is a sign-changing continuous function. We show the existence andmultiplicity of non-negative solutions of $(P_{\la})$ with respect to theparameter $\la$, 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.

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