A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function

Abstract

In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: $$ \quad \left\{\begin{array}{lr}\ds \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dxdy = \la h(x)|u|^{q-1}u+ b(x)|u|^{r-1} u \; \text{in}\; \Om \quad \quad \quad \quad u \geq 0 \; \mbox{in}\; \Om,\quad u\in W^{\al,p}(\mb R^n), \quad \quad\quad \quad\quad u =0\quad\quad \text{on} \quad \mb R^n\setminus \Om \end{array} \right. $$ where $\Om$ is a bounded domain in $\mb R^n$, $p\geq 2$, $n> p\al$, $\al\in(0,1)$, $0< q<p-1 <r < \frac{np}{n-ps}-1$, $\la>0$ and $h$, $b$ are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists $\la_0$ such that for $\la\in (0,\la_0)$, it has at least two solutions.