BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS
Teresa D’Aprile, Juncheng Wei · Journal of the London Mathematical Society · 2006
We study concentration phenomena for the system ε 2 Δ υ − υ − δ φ υ + γ υ p = 0 , Δ φ + δ υ 2 = 0 in the unit ball B1 of ℝ3 with Dirichlet boundary conditions. Here ɛ, δ, γ > 0 and p > 1. We prove the existence of positive radial solutions (υɛ, φɛ) such that υɛ concentrates at a distance (ɛ/2)|log ɛ| away from the boundary ∂ B1 as the parameter ɛ tends to 0. The approach is based on a combination of Lyapunov–Schmidt reduction procedure together with a variational method.