Another proof of a Lions type existence result

TL;DR

This paper proves the existence of positive solutions for a nonlinear elliptic equation with critical Sobolev growth under Lions's condition, using variational methods and Palais--Smale sequences.

Contribution

It provides a new proof of Lions's existence result for a class of nonlinear elliptic equations with critical growth.

Findings

Existence of at least one positive solution established

Construction of a compact Palais--Smale sequence

Application of variational methods to critical Sobolev problems

Abstract

This paper concerns a nonlinear elliptic equation involving a critical Sobolev growth and a lower-order term. Under a Lions's condition, we prove the existence of at least one positive solution. Our approach consists in constructing a relatively compact Palais--Smale sequence for the associated variational problem.