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New type of solutions for the critical Lane-Emden system

Wenjing Chen, Xiaomeng Huang

Abstract

In this paper, we consider the critical Lane-Emden system \begin{align*} \begin{cases} -Δu=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -Δv=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where $N\geq 5$, $p,q\in (1,\infty)$ with $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. Under suitable conditions on $K_1(y)$ and $K_2(y)$, we construct a new family of solutions to this system, which are centred at points lying on the top and the bottom circles of a cylinder.

New type of solutions for the critical Lane-Emden system

Abstract

In this paper, we consider the critical Lane-Emden system \begin{align*} \begin{cases} -Δu=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -Δv=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where , with , and are positive radial potentials. Under suitable conditions on and , we construct a new family of solutions to this system, which are centred at points lying on the top and the bottom circles of a cylinder.
Paper Structure (4 sections, 16 theorems, 311 equations)

This paper contains 4 sections, 16 theorems, 311 equations.

Table of Contents

  1. Introduction and statement of main result
  2. Finite-dimensional reduction
  3. Proof of the existence result
  4. Appendix

Key Result

Theorem 1.1

Suppose that $N\geq 5$ and $p$, $q$, $m$ satisfy (pq), (p+), (q+), (m0) and (m+) respectively. If $K_i(r)$ satisfies $(K_i)$, then problem (le) has infinitely many non-radial solutions.

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 20 more