On local compactness in quasilinear elliptic problems

TL;DR

This paper derives a general formula for the critical level of Palais-Smale sequences in nonlinear elliptic problems with critical nonlinearities, addressing compactness issues in such problems.

Contribution

It provides a natural, general formula for the critical level in a broad class of nonlinear elliptic critical problems, extending previous work.

Findings

Derived a formula for the critical level in elliptic problems

Constructed Palais-Smale sequences demonstrating sharpness

Addressed compactness issues in nonlinear elliptic problems

Abstract

One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Br\'ezis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact.