Schrödinger type equations with asymptotically linear nonlinearities

Abstract

We study the nonlinear Schrödinger type equation $$-\Delta u~+~(\lambda g(x)~+~1)u~=~f(u)$$ on the whole space ${{\mathbf R}^N}$. The nonlinearity $f$ is assumed to be asymptotically linear and $g(x)\geq 0$ has a potential well. We do not assume a limit for $g(x)$ as $|x|\to\infty$. Using variational techniques, we prove the existence of a positive solution for $\lambda$ large. In the case where $f$ is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as $\lambda\to\infty$ is also considered.