Layered solutions for a semilinear elliptic system in a ball

We consider the following system of Schrödinger–Poisson equations in the unit ball B1 of : with the Dirichlet boundary conditions v==0 on ∂B1. Here ε, ω, γ>0, v, , . We exhibit a family of positive radially symmetric solutions (vε,ε) such that vε concentrates on a sphere in the interior of B1 as ε→0+. Our approach relies upon a finite-dimensional reduction by the Lyapunov–Schmidt method which allows us to reduce the problem to maximizing a one-dimensional functional. The solution of such reduced problem also provides the radius of the concentration sphere.