On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

Abstract

We study the existence of nontrivial solutions for the problem Δu=u, in a bounded smooth domain Ω⊂ℝℕ, with a semilinear boundary condition given by ∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, where W is a potential changing sign, g has a superlinear growth condition, and the parameter λ∈]0,λ1];λ1 is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.