Multiple solutions to boundary value problems for semilinear elliptic equations

In this article, we study the multiplicity of weak solutions to the boundary value problem

-Δu = ƒ(x, u) + g(x, u) in Ω,

u = 0 on ∂Ω,

where Ω is a bounded domain with smooth boundary in ℝ^N (N > 2), ƒ(x, ξ) is odd in ξ and g is a perturbation term. Under some growth conditions on ƒ and g, we show that there are infinitely many solutions. Here we do not require that ƒ be continuous or satisfy the Ambrosetti-Rabinowitz (AR) condition. The conditions assumed here are not implied by the ones in [3, 15]. We use the perturbation method Rabinowitz combined with estimating the asymptotic behavior of eigenvalues for Schrödinger's equation.