Schrödinger type equations with asymptotically linear nonlinearities (Q1413569)

The authors consider existence of solutions for the following nonlinear Schrödinger type equation \[ -\Delta u+(\lambda g(x)+1)u=f(u)\quad \text{in } \mathbb R^n \] which satisfy \(u(x)\to 0\) as \(|x|\to \infty.\) The nonlinearity \(f\) is assumed to be asymptotically linear and \(g(x)\geq 0\) has a potential well. Using variational techniques the authors prove existence of a positive solution for \(\lambda\) large. Multiple pairs of solutions are obtained in the case \(f\) is odd. The limiting behavior of solutions as \(\lambda\to \infty\) is also considered.