Multiple Boundary Peak Solution for Critical Elliptic System with Neumann Boundary

2024/02/26 by Yuxia Guo, Guo, Yuxia, Shengyu Wu +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Differential Equations and Numerical Methods

paper · pdf · doi:10.48550/arxiv.2402.16489

Abstract

We consider the following elliptic system with Neumann boundary: \begincases -Δu + μu=v^p, amp;\hboxin Ω,
-Δv + μv=u^q, amp;\hboxin Ω,
(∂ u)/(∂ n) = (∂ v)/(∂ n) = 0, amp;\hboxon ∂Ω,
ugt;0,vgt;0, amp;\hboxin Ω, \endcases where Ω⊂ ℝ^N is a smooth bounded domain, μ is a positive constant and (p,q) lies in the critical hyperbola: \dfrac1p+1 + \dfrac1q+1 =\dfracN-2N. 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 ∂ Ω. Our results show that the geometry of the boundary ∂Ω, especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.

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