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} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, &\hbox{on } \partial\Omega, \\u>0,v>0, &\hbox{in } \Omega, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $\mu$ 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 $\partial \Omega$. Our results show that the geometry of the boundary $\partial\Omega,$ especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.