Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 79, pp. 1-23.

Title: A global compactness result for quasilinear elliptic problems with critical 
Sobolev nonlinearities and Hardy potentials on R^N

Authors: Lingyu Jin (South China Agricultural Univ., Guangzhou, China)
         Suting Wei (South China Agricultural Univ., Guangzhou, China)
  
Abstract: 
In this article, we study the elliptic equation with critical Sobolev 
nonlinearity and Hardy potentials
$$
(-\Delta)_p u+a(x) |u|^{p-1}u-\mu\frac{|u|^{p-1}u}{|x|^p}
=|u|^{p^*-2}u+f(x,u),\quad u \in W^{1,p}(\mathbb{R}^N),
$$
where  $0<\mu<\min\{\frac{(N-p)^p}{p^p}, \frac{N^{p-1}(N-p^2)}{p^p}\}$, 
$p^*=\frac{Np}{N-p}$ is the critical Sobolev exponent. 
Through a compactness analysis of the  associated  functional operator,
we obtain the existence of positive solutions under certain assumptions on 
$a(x)$ and $f(x,u)$.

Submitted April 5, 2024. Published December 03, 2024.
Math Subject Classifications: 35J10, 35J20, 35J60.
Key Words: p-Laplacian; compactness; positive solution; unbounded domain; Sobolev nonlinearity.