Multiple sign-changing solutions to semilinear elliptic resonant problems

In this paper, we use the topological degree theory and the critical groups to investigate the elliptic equation −Δu=f(x,u)−Δu=f(x,u) in ΩΩ and subject to u=0u=0 on ∂Ω∂Ω, and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions. If this problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions.