On an elliptic equation with concave and convex nonlinearities

Abstract

We study the semilinear elliptic equation − Δ u = λ | u | q − 2 u + μ | u | p − 2 u - \Delta u = \lambda |u{|^{q - 2}}u + \mu |u{|^{p - 2}}u in an open bounded domain Ω ⊂ R N \Omega \subset {\mathbb {R}^N} with Dirichlet boundary conditions; here 1 > q > 2 > p > 2 ∗ 1 > q > 2 > p > {2^ \ast } . Using variational methods we show that for λ > 0 \lambda > 0 and μ ∈ R \mu \in \mathbb {R} arbitrary there exists a sequence ( v k ) ({v_k}) of solutions with negative energy converging to 0 as k → ∞ k \to \infty . Moreover, for μ > 0 \mu > 0 and λ \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.