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)&-λh(x)|x|^{-2(1+a)}u=μf(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N, \end{align*} where $h(x)\ge 0$, $f(x)$ is a continuous function which may change sign, $\lambda ,\mu$ are positive real parameters and $1<q<2$, $4<p=2N/[N+2(b-a)-2]$, $0\le a<b<a+1<N/2$, $N\ge 3$. Here $$L(u)=-M\left({\int }_{{\mathbb{R}}^N}|x{|}^{-2a}|\nabla u{|}^{2}dx\right)\mathrm{div}(|x{|}^{-2a}\nabla u)$$ 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 $\mu$.