On an elliptic equation with concave and convex nonlinearities

Abstract:

We study the semilinear elliptic equation $- \Delta u = \lambda |u{|^{q - 2}}u + \mu |u{|^{p - 2}}u$ in an open bounded domain $\Omega \subset {\mathbb {R}^N}$ with Dirichlet boundary conditions; here $1 < q < 2 < p < {2^ \ast }$. Using variational methods we show that for $\lambda > 0$ and $\mu \in \mathbb {R}$ arbitrary there exists a sequence $({v_k})$ of solutions with negative energy converging to 0 as $k \to \infty$. Moreover, for $\mu > 0$ and $\lambda$ arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brézis 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.