An asymmetric superlinear elliptic problem at resonance

In this article, we study the existence and multiplicity of solutions to the problem {−Δu=g(x,u),  in  Ω;u=0,  on  ∂Ω, where ΩΩ is a bounded domain in RNRN(N⩾2)(N⩾2) with smooth boundary, and g:Ω¯×R→R is a differentiable function. We will assume that g(x,s)g(x,s) has a resonant behavior for large negative values of ss and that a Landesman–Lazer type condition is satisfied. We also assume that g(x,s)g(x,s) is superlinear, but subcritical, for large positive values of ss. We prove the existence and multiplicity of solutions for problem (1.1) by using minimax methods and infinite-dimensional Morse theory.