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Multiple Boundary Peak Solution for Critical Elliptic System with Neumann Boundary

Yuxia Guo, Shengyu Wu, TingFeng Yuan

Abstract

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -Δu + μu=v^p, &\hbox{in } Ω, \\-Δv + μv=u^q, &\hbox{in } Ω, \\\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, &\hbox{on } \partialΩ, \\u>0,v>0, &\hbox{in } Ω, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $μ$ is a positive constant and $(p,q)$ lies in the critical hyperbola: $$ \dfrac{1}{p+1} + \dfrac{1}{q+1} =\dfrac{N-2}{N}. $$ By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary $\partial Ω$. Our results show that the geometry of the boundary $\partialΩ,$ especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.

Multiple Boundary Peak Solution for Critical Elliptic System with Neumann Boundary

Abstract

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -Δu + μu=v^p, &\hbox{in } Ω, \\-Δv + μv=u^q, &\hbox{in } Ω, \\\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, &\hbox{on } \partialΩ, \\u>0,v>0, &\hbox{in } Ω, \end{cases} \end{equation} where is a smooth bounded domain, is a positive constant and lies in the critical hyperbola: By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary . Our results show that the geometry of the boundary especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.
Paper Structure (6 sections, 28 theorems, 262 equations)

This paper contains 6 sections, 28 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Finite Dimensional Reduction
  3. Energy estimates
  4. Proof of Theorem 1.1
  5. Essential Results
  6. Basic Estimate

Key Result

Theorem 1.1

Suppose $N \geq 5$ and $p$ satisfies condition (A). Besides, $\Omega$ is a smooth bounded domain satisfying (H1), (H2), (H3), and $\mu$ is a fixed positive constant. Then there exists $k_0>0$, such that for any $k > k_0$, problem equ-2 admits a solution with the following form: where $\omega_1$ and $\omega_2$ are error term that satisfies Neumann condition.

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 39 more