ON AN ELLIPTIC EQUATION WITH CONCAVE AND CONVEX NONLINEARITIES

We study the semilinear elliptic equation -Delta u=lambda\u\(q-2)u+mu\u\(p-2)u in an open bounded domain Omega subset of R(N) with Dirichlet boundary conditions; here 1 < q < 2 < p < 2*. Using variational methods we show that for lambda > 0 and mu is an element of R arbitrary there exists a sequence (upsilon(k)) of solutions with negative energy converging to 0 as k --> infinity. Moreover, for mu > 0 and lambda arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brezis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.