Existence of infinitely many solutions for a p-Kirchhoff problem in RN

Abstract

AbstractWe consider the existence of multiple solutions of the following singular nonlocal elliptic problem: $$\begin{aligned} \textstyle\begin{cases} -M(\int _{\mathbb{R} ^{N}}{ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p}})\operatorname{div}( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= h(x) \vert u \vert ^{r-2}u+H(x) \vert u \vert ^{q-2}u, \\ u(x)\rightarrow 0 \quad \text{as } \vert x \vert \rightarrow \infty , \end{cases}\displaystyle \end{aligned}$$ {−M(∫RN|x|−ap|∇u|p)div(|x|−ap|∇u|p−2∇u)=h(x)|u|r−2u+H(x)|u|q−2u,u(x)→0as |x|→∞, where $x\in \mathbb{R} ^{N}$x∈RN, and $M(t)=\alpha +\beta t$M(t)=α+βt. By the variational method we prove that the problem has infinitely many solutions when some conditions are fulfilled.