An Eigenvalue Problem for Nonlinear Schrödinger-Poisson System with Steep Potential Well

In this paper, we study an eigenvalue problem for SchrodingerPoisson system with indefinite nonlinearity and potential well as follows: (-Delta u + mu V (x)u + K (x)phi u = lambda f(x)u + g(x)vertical bar u vertical bar(p-2)u in R-3, -Delta phi = K (x)u(2) in R-3, where 4 <= p < 6, the parameters mu, lambda > 0, V is an element of C (R-3) is a potential well with the bottom (Omega) over bar := {x is an element of R-3 : V (x) = 0}, and the functions f and g are allowed to be sign-changing. By establishing an approximate estimate between the positive principal eigenvalue of -Delta u + mu V (x)u = lambda f(x)u in R-3 and the positive principal eigenvalue lambda(1)(f(Omega)) of -Delta u = lambda f(Omega)u in Omega, we prove that at least a positive solution exists in 0 < lambda <= lambda(1) (f(Omega)) while at least two positive solutions exist in lambda > lambda(1) (f(Omega)) and near lambda(1)(f(Omega)), where f(Omega) := f vertical bar(Omega). The results are obtained via variational method and steep potential. Furthermore, we also study the concentration of solutions as mu -> infinity and the decay rate of solutions at infinity.