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Elliptic systems in Orlicz-Sobolev spaces with critical sources in bounded domains

Pablo Ochoa

TL;DR

The paper addresses finding nontrivial weak solutions for elliptic systems with critical nonlinearities in Orlicz-Sobolev spaces, a setting that accommodates nonlinearities beyond polynomial growth. It combines a Mountain Pass theorem without the Palais–Smale condition with Lions' concentration-compactness in Orlicz spaces to overcome lack of compactness. Under general growth and structural assumptions on the $g_i$-Laplacians and the nonlinearities $F$ and $H$, it establishes existence of nontrivial solutions for sufficiently large $\lambda$, including a variant that relaxes the nonnegativity of $H$. This advances the theory by extending critical-growth results to broad Orlicz-Sobolev contexts and demonstrates the viability of CCP techniques in these spaces for elliptic systems.

Abstract

In this paper, we show the existence of non-trivial solutions to very general elliptic systems with critical non-linearities in the sense of embeddings in Orlicz-Sobolev spaces. This allows to consider non-linearities which do not have polynomial growth. To achieve the existence, we combine a Mountain Pass Theorem without the Palais-Smale condition with the second Concentration Compactness Principle of Lions in Orlicz-Sobolev spaces.

Elliptic systems in Orlicz-Sobolev spaces with critical sources in bounded domains

TL;DR

The paper addresses finding nontrivial weak solutions for elliptic systems with critical nonlinearities in Orlicz-Sobolev spaces, a setting that accommodates nonlinearities beyond polynomial growth. It combines a Mountain Pass theorem without the Palais–Smale condition with Lions' concentration-compactness in Orlicz spaces to overcome lack of compactness. Under general growth and structural assumptions on the -Laplacians and the nonlinearities and , it establishes existence of nontrivial solutions for sufficiently large , including a variant that relaxes the nonnegativity of . This advances the theory by extending critical-growth results to broad Orlicz-Sobolev contexts and demonstrates the viability of CCP techniques in these spaces for elliptic systems.

Abstract

In this paper, we show the existence of non-trivial solutions to very general elliptic systems with critical non-linearities in the sense of embeddings in Orlicz-Sobolev spaces. This allows to consider non-linearities which do not have polynomial growth. To achieve the existence, we combine a Mountain Pass Theorem without the Palais-Smale condition with the second Concentration Compactness Principle of Lions in Orlicz-Sobolev spaces.

Paper Structure

This paper contains 10 sections, 15 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. N-functions and basic properties
  4. Main assumptions on the data
  5. Preliminary results
  6. Proof of Theorem \ref{['main theorem']}
  7. Geometry of the energy functional and boundedness of the Palais-Smale sequence
  8. Analysis of the PS-sequence via the Concentration Compactness Principle
  9. Final argument
  10. Proof of Theorem \ref{['main theorem 2']}

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with the cone property. Consider the elliptic system main system, where the $N$-functions $G_1$ and $G_2$ satisfy G1, GG, GGG and exponents. Regarding the data $F$ and $H$, we assume that they verify the conditions $(F1)-(F3)$ and $(H1)-(H3)$, re

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 3.1
  • Example 3.2
  • ...and 19 more