Existence and multiplicity for a superlinear elliptic problem under a non-quadradicity condition at infinity

Abstract

In this article, we study the existence and multiplicity of solutions of the boundary-value problem -Δu = ƒ(x, u), in Ω, u = 0, on ∂Ω where ∆ denotes the N-dimensional Laplacian, Ω is a bounded domain with smooth boundary, ∂Ω, in ℝN (N ≯ 3), and ƒ is a continuous function having subcritical growth in the second variable. Using infinite-dimensional Morse theory, we extended the results of Furtado and Silva [9] by proving the existence of a second nontrivial solution under a non-quadradicity condition at infinity on the non-linearity. Assuming more regularity on the non-linearity ƒ, we are able to prove the existence of at least three nontrivial solutions.