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 $\Delta u=u$, in a bounded smooth domain $\Omega \subset {ℝ}^{\mathbb{N}}$, with a semilinear boundary condition given by $\partial u/\partial \nu =\lambda u-W\left(x\right)g\left(u\right)$, on the boundary of the domain, where $W$ is a potential changing sign, $g$ has a superlinear growth condition, and the parameter $\lambda \in \left]0,{\lambda }_1\right];{\lambda }_1$ is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.