Nehari manifold for non-local elliptic operator with concavea€“convex nonlinearities and sign-changing weight functions

摘要

In this article, we study the existence and multiplicity of non-negative solutions of the following p-fractional equation:egin{equation*}left{egin{matrix}-2 {displaystyleint}_{mathbb{R}^n} frac{|u(y) - u (x)|^{p-2} (u(y)-u(x))}{|x-y|^{n+pe???}} dy = e??? h (x) |u|^{q-1} u + b (x)|u|^{r-1} u ext{ in } e??o,u = 0 quad ext{ in } mathbb{R}^n setminus e??o, quad u in W^{e???,p} (mathbb{R}^n)end{matrix} ight.end{equation*}where e??o is a bounded domain in $mathbb{R}^n$ with continuous boundary, $p a‰￥ 2$, $n p e???$, $e??? in (0,1)$, $0 q p -1 r p^* - 1$ with $p^* = np (n -pe???)^{-1}$, $e??? 0$ and $h, b$ are signchanging continuous functions. We show the existence and multiplicity of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists e???0 such that for $e??? in (0, e???_0)$, it has at least two non-negative solutions.