W01,1(Ω)-solutions for a degenerate double phase type operator in some borderline cases

Abstract

In this paper we study the existence of W₀^1,1(Ω)-solutions of the nonlinear problems which involves in its principal part the p-laplacian operator and a degenerate additional term that has a polynomial growth with respect to the gradient. The simplest model is −div (a(x)|∇u|^p−2∇u) -div (|u|^{( r−1)q+1}|∇u|^q−2∇u) = f  in Ω, and u = 0 on ∂Ω, where Ω is a bounded open subset of R^N(N>2), 1 < q ≤ p < N,  r > q-1/q and f is a function with poor summability.