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Bounded Solutions of Lane-Emden-Fowler system in 2D Exterior Domains

Dragos-Patru Covei

Abstract

This research investigates the Lane-Emden-Fowler sublinear system in two-dimensional exterior domains, motivated by the corresponding scalar cases studied by Constantin (1997) and Yin (2004). Utilizing Liouville transformation, Schauder's fixed point theorem and sub-supersolution method, we establish the existence of bounded positive solutions under suitable conditions on the functions involved. As citations in the papers by Constantin (1997) and Yin (2004) show, this work is the first to consider the system case.

Bounded Solutions of Lane-Emden-Fowler system in 2D Exterior Domains

Abstract

This research investigates the Lane-Emden-Fowler sublinear system in two-dimensional exterior domains, motivated by the corresponding scalar cases studied by Constantin (1997) and Yin (2004). Utilizing Liouville transformation, Schauder's fixed point theorem and sub-supersolution method, we establish the existence of bounded positive solutions under suitable conditions on the functions involved. As citations in the papers by Constantin (1997) and Yin (2004) show, this work is the first to consider the system case.

Paper Structure

This paper contains 8 sections, 4 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Proof of the main results
  3. Proof of Theorem \ref{['lef']}.
  4. Proof of Theorem \ref{['cons']}.
  5. Proof of Lemma \ref{['subsuper']}.
  6. Proof of Theorem \ref{['cons']} completed.
  7. Proof of the additional result
  8. Proof of Theorem \ref{['thm:special01']}

Key Result

Theorem 1.1

Let $c\in \left[ 1,\infty \right)$. Assume that the functions $p$ and $q$ are positive radial continuous in $G_{A}$. If, in addition then, there exists $B_{c}\geq A$ such that the system (s1) has a bounded positive radial solution and $\left( u,v\right)$ satisfies the system (s1) at every point $x\in G_{B_{c}}$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Remark 3.1