Weighted Sobolev spaces and Morse estimates for quasilinear elliptic equations

Abstract

We establish critical groups estimates for the weak solutions of −Δpu=f(x,u) in Ω and u=0 on ∂Ω via Morse index, where Ω is a bounded domain, f∈C1(Ω‾×R) and f(x,s)>0 for all x∈Ω‾, s>0 and f(x,s)=0 for all x∈Ω‾, s≤0. The proof relies on new uniform Sobolev inequalities for approximating problems. We also prove critical groups estimates when Ω is the ball or the annulus and f is a sign changing function.