Multiple Nontrivial Solutions of Semilinear Elliptic Equations and Their Homotopy Indexes

In this paper, we consider the semilinear elliptic equation −Δu(x) = g(u(x)), x Ωu(x) = 0, x ∂Ω, where Ω Rn is a bounded domain with smooth boundary ∂Ω and g : R → R is of class C2. It is clear that solutions of this problem are equilibria of the local semiflow πg generated by solutions of the semilinear parabolic equation u(t)(t, x) = Δu(t, x) + g(u(t, x)), t ≥ 0, x Ωu(t, x) = 0, t ≥ 0 x ∂Ω. We study the existence of multiple nontrivial solutions of the above elliptic equation and their homotopy indices. The homotopy index theory is an extension of Conley′s index theory to noncompact spaces due to Rybakowski and gives some information about the local semiflow πg.