Weighted Sobolev spaces and Morse estimates for quasilinear elliptic equations

We establish critical groups estimates for the weak solutions of − Δ_p u = f(x, u) in Ω and u = 0 on ∂Ω via Morse index, where Ω is a bounded domain, f ∈ C^1(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.