Caffarelli-Kohn-Nirenberg type problems with Berestycki-Lions type nonlinearities

Abstract

In this article we use a Palais-Smale sequence satisfying a  property related to Pohozaev identity to show the existence of  solution for the elliptic Caffarelli-Kohn-Nirenberg type problems  $$ -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u) + |x|^{-bp^{*}}|u|^{p-2}u=   |x|^{-bp^{*}}h(u) \quad \text{in }\mathbb{R}^N $$ and $$ -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u) =  |x|^{-bp^{*}} f(u) \quad \text{in }\mathbb{R}^N, $$ where \(1<p<N\), \(0\leq a< \frac{N-p}{p^{*}}\), \(a< b\leq a+1\),   \(p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}\) and \(d=1+a-b\). and \(h\) and  \(f\) are  continuous functions that satisfy hypotheses considered by  Berestycki and Lions in [7].  For more information see https://ejde.math.txstate.edu/Volumes/2024/44/abstr.html