Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 44, pp. 1-12.

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

Authors: Giovany M. Figueiredo (Univ. de Brasilia, Brazil)
         George Kiametis (Univ. de Brasilia, Brazil)

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 
$$
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)
+ |x|^{-bp^{*}}|u|^{p-2}u=   |x|^{-bp^{*}}h(u) \quad \text{in }R^N
$$
and
$$
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)
=  |x|^{-bp^{*}} f(u) \quad \text{in }R^N,
$$
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].

Submitted January 15, 2024. Published August 12, 2024.
Math Subject Classifications: 35B38, 35J35, 35J92.
Key Words: Caffarelli-Kohn-Nirenberg type problems; Nehari manifold.