Application of a profile decomposition theorem to elliptic equations with critical growth

TL;DR

This paper develops new variational methods using profile decomposition to prove the existence of ground state solutions for elliptic equations with critical growth in an asymptotically periodic setting, accommodating complex nonlinearities.

Contribution

It introduces a novel approach applying profile decomposition directly to Sobolev sequences, handling general nonlinearities without classical conditions, and establishing ground state existence under broad assumptions.

Findings

Proved existence of ground states for elliptic equations with critical growth.

Extended applicability to nonlinearities with oscillatory behavior.

Removed need for classical conditions like Ambrosetti-Rabinowitz.

Abstract

This paper introduces new variational methods centered on the direct application of a profile decomposition theorem for bounded sequences in Sobolev spaces. We employ these methods to prove the existence of ground state solutions for a class of semilinear elliptic equations in $\mathbb{R}^N$ with critical Sobolev growth, set in an asymptotically periodic framework where the coefficients converge to periodic functions at infinity. Our approach successfully addresses highly general nonlinearities, including a subcritical term that does not need to satisfy the classical Ambrosetti-Rabinowitz condition and a critical term that extends far beyond the standard pure power assumption to include functions with oscillatory behavior. We prove the existence of ground states under two alternative conditions: either a strict energy gap between the minimax levels of the original and asymptotic problems or a direct energy comparison between the associated functionals. Some restrictive assumptions, such as specific decay rates for the coefficients or monotonicity properties of the nonlinearities, are not required in our results.