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

Diego Ferraz

TL;DR

This work tackles the existence of ground state solutions for elliptic equations in $\mathbb{R}^N$ with critical Sobolev growth under asymptotically periodic coefficients, by developing a direct variational approach based on a profile decomposition theorem. It handles highly general nonlinearities, including oscillatory self-similar critical terms and subcritical perturbations not satisfying the Ambrosetti–Rabinowitz condition, and proves ground states under either a strict energy-gap condition between the original and asymptotic problems or via a direct energy comparison. The analysis is carried out in two phases: first a semi-autonomous framework with $k(x,u)=b(x)g(u)$, then an extension to fully nonautonomous problems, using profile decomposition to control loss of compactness and to decompose energy across limit problems. The results significantly broaden the scope of variational methods for noncoercive problems, eliminating many decay-rate and monotonicity restrictions and accommodating oscillatory, self-similar nonlinearities in both autonomous and nonautonomous settings.

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.

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

TL;DR

This work tackles the existence of ground state solutions for elliptic equations in with critical Sobolev growth under asymptotically periodic coefficients, by developing a direct variational approach based on a profile decomposition theorem. It handles highly general nonlinearities, including oscillatory self-similar critical terms and subcritical perturbations not satisfying the Ambrosetti–Rabinowitz condition, and proves ground states under either a strict energy-gap condition between the original and asymptotic problems or via a direct energy comparison. The analysis is carried out in two phases: first a semi-autonomous framework with , then an extension to fully nonautonomous problems, using profile decomposition to control loss of compactness and to decompose energy across limit problems. The results significantly broaden the scope of variational methods for noncoercive problems, eliminating many decay-rate and monotonicity restrictions and accommodating oscillatory, self-similar nonlinearities in both autonomous and nonautonomous settings.

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 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.
Paper Structure (22 sections, 34 theorems, 161 equations)

This paper contains 22 sections, 34 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Standard hypotheses
  3. First case: Semi-autonomous critical nonlinearity
  4. Second case: The fully nonautonomous problem
  5. Remarks on the assumptions
  6. Outline
  7. Preliminaries
  8. Profile decomposition for bounded sequences
  9. Existence of a bounded Palais-Smale sequence
  10. Behavior of the functional under profile decomposition
  11. $\mathbb{Z}^N$--Periodic case: Proof of Theorem \ref{['teo_periodic']}
  12. Asymptotic $\mathbb{Z}^N$--periodic case: Proof of Theorem \ref{['teo_compact']}
  13. Existence of ground states: Proof of Theorem \ref{['teo_gs']}
  14. Existence of ground states for the fully nonautonomous case: Proof of Theorems \ref{['teo_periodic2']}--\ref{['teo_gs2']}
  15. Boundedness of Palais-Smale sequences
  16. ...and 7 more sections

Key Result

Theorem 1

MR2465979 Let $(u_k) \subset D^{1,2}(\mathbb{R}^N)$ be a bounded sequence in the standard $D^{1,2}(\mathbb{R}^N)$--norm and $\gamma > 1.$ There exist $(w^{(n)})_{n \in \mathbb{N}_\ast} \subset D^{1,2}(\mathbb{R}^N),$$(y^{(n)}_k)_{k \in \mathbb{N}}\subset \mathbb{Z}^N,$$(j_k^{(n)})_{k \in \mathbb{N}} and the series in tinta4 converges uniformly in $k.$ Furthermore, $1\in \mathbb{N}_0,$$y_k^{(1)} =

Theorems & Definitions (64)

  • Theorem 1
  • Theorem 1.1: $\mathbb{Z}^N$--periodic case
  • Theorem 1.2: Compactness
  • Theorem 1.3: Ground state
  • Theorem 1.4: $\mathbb{Z}^N$--periodic case
  • Theorem 1.5: Compactness
  • Theorem 1.6: Ground state
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 54 more