Weighted Sobolev spaces and Morse estimates for quasilinear elliptic equations

We establish critical groups estimates for the weak solutions of −Δp u =f(x,u) in Ω and u =0 on ∂Ω via Morse index, where Ωis 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 fis a sign changing function.