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

In this paper we study the following class of nonlocal problem involving Caffarelli-Kohn-Nirenberg type critical growth $$ L(u)-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\quad \text{in } \mathbb R^N, $$% 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} \big(|x|^{-2a}\nabla u\big) $M\colon R^+_0\to\mathbb R^+_0$ exactly Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta> 0$. The above has double lack compactness, firstly because non-compactness embedding secondly due to inclusion map $$u\mapsto \int_{\mathbb R^N}h(x)|x|^{-2(a+1)}|u|^2dx,$$ as domain in consideration unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, prove existence at least two solutions for suitable choices $\lambda$ $\mu$.