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Decay estimates and symmetry of finite energy solutions to elliptic systems in R^n

Jérôme Vétois

Abstract

We study a notion of finite energy solutions to elliptic systems with power nonlinearities in R^n. We establish sharp pointwise decay estimates for positive and sign-changing solutions. By using these estimates, we obtain symmetry results when the solutions are positive.

Decay estimates and symmetry of finite energy solutions to elliptic systems in R^n

Abstract

We study a notion of finite energy solutions to elliptic systems with power nonlinearities in R^n. We establish sharp pointwise decay estimates for positive and sign-changing solutions. By using these estimates, we obtain symmetry results when the solutions are positive.

Paper Structure

This paper contains 2 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction and main results
  2. Regularity and integrability results

Key Result

Theorem 1.1

Assume that Eq2 and Eq5 hold true. Let $$u,v$\in L^a$R^n$\times L^b$R^n$$ be a solution of Eq1 such that Eq8 and Eq9 hold true. Then $u,v\in C^2$R^n$$ and there exists a constant $C_0$ such that where Moreover, if $u,v\ge0$ in $\mathbb{R}^n$, then either $u\equiv v\equiv0$ in $\mathbb{R}^n$ or there exists a constant $C_1>0$ such that

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4