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Blowing-up solutions to a critical 4D Neumann system in a competitive regime

Qing Guo, Angela Pistoia, Shixin Wen

Abstract

We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -Δu_1 + λu_1 = u_1^{3} -βu_1u_2^2 & \text{in } Ω, -Δu_2 + λu_2 = u_2^{3} -βu_1^2u_2 & \text{in } Ω, \frac{\partial u_1}{\partialν} = \frac{\partial u_2}{\partialν} = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} when $λ>0$ is sufficiently large in a competitive regime (i.e. $ β>0$) and in a domain $Ω\subset\mathbb R^4$ with smooth protrusions.

Blowing-up solutions to a critical 4D Neumann system in a competitive regime

Abstract

We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -Δu_1 + λu_1 = u_1^{3} -βu_1u_2^2 & \text{in } Ω, -Δu_2 + λu_2 = u_2^{3} -βu_1^2u_2 & \text{in } Ω, \frac{\partial u_1}{\partialν} = \frac{\partial u_2}{\partialν} = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} when is sufficiently large in a competitive regime (i.e. ) and in a domain with smooth protrusions.

Paper Structure

This paper contains 8 sections, 6 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ansatz
  4. Setting of the problem
  5. Proof of the Theorem (\ref{['xthmy']})
  6. Properties of the function $W$
  7. Solving problem (\ref{['vec-1']})
  8. Proof of Proposition \ref{['energy-esti']}

Key Result

Theorem 1.1

Assume that the mean curvature function $H$ admits two distinct strict local maximum points $\xi_i^*$ with $H(\xi_i^*)>0$ for $i=1,2$. For any $\beta>0$, there exists $\lambda_0>0$ such that for any $\lambda \in (\lambda_0, +\infty)$, system n-4 admits a solution $(u_{1\lambda}, u_{2\lambda})$ such

Figures (1)

  • Figure :

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 1.1
  • ...and 8 more