Morse index and multiple solutions for the asymptotically linear Schrödinger type equation

We study the nonlinear problem −Δu+V(x)u=f(x,u), x∈RN, where the Schrödinger type operator −Δ+V is semibounded and has a finite number of eigenvalues below the infimum of the essential spectrum. We classify the linear Schrödinger equation, prove the (semi-)continuity and monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Schrödinger equations, based on an application of the classical Ambrosetti-Rabinowitz symmetric Mountain-Pass Theorem.