The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

Abstract

In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth \begin{align*} L(u)&-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N, \end{align*} where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1<q<2$, $4< p=2N/[N+2(b-a)-2]$, $0\leq a<b<a+1<N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div}(|x|^{-2a}\nabla u) $$ and the function $M:\mathbb R^+\cup \{0\} \to\mathbb R^+$ is exactly as in the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta>0$. Using the idea {of the constrained minimization on} Nehari manifold we show the existence of at least two positive solutions for suitable choices of $\lambda$ and $\mu$.