Multiple Boundary Peak Solution for Critical Elliptic System with Neumann Boundary

We consider the following elliptic system with Neumann boundary: equation cases - u + u=vᵖ, &in, \\- v + v=uq, &in, \\ u n = v n = 0, &on, \>0, v>0, &in, cases equation where RN is a smooth bounded domain, is a positive constant and (p, q) lies in the critical hyperbola: 1p+1 + 1q+1 =N-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. 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.