On some indefinite and non-powerlike elliptic equations

We consider a class of elliptic problems for the operator −Δ+V−Δ+V, acting on H01(Ω), where ΩΩ is a bounded domain. The main feature here is the indefinite behavior of both the operator and the nonlinearity. Based on the global geometry of the energy functional, we prove the existence of two local minimizers and a mountain-pass critical point for a concave–convex type nonlinearity. Subcritical and critical growth are allowed on the nonlinearity.