Compactness results for Schrödinger equations with asymptotically linear terms

We study the nonlinear problem −Δu+V(x)=f(x,u), , limx→∞u(x)=0, where the Schrödinger operator −Δ+V is semibounded and the nonlinearity f is linearly bounded. We establish compactness of Palais–Smale sequences and Cerami sequences for the associated energy functional under general spectral-theoretic assumptions. Applying these results, we obtain existence of three nontrivial solutions if the energy functional has a mountain-pass geometry.