A quasilinear problem with fast growing gradient

Original · EN

In this paper we consider the following Dirichlet problem for the p-Laplacian in the positive parameters λ and β: [array [c]rcll% -Δₚu & = & λh(x,u)+βf(x,u,∇ u) & inΩu & = & 0 & on∂Ω, array.] where h,f are continuous nonlinearities satisfying 0≤ω₁(x)uq⁻¹≤ h(x,u)≤ω₂(x)uq⁻¹ with 1<q<p and 0≤ f(x,u,v)≤ω₃(x)uᵃ|v|ᵇ, with a,b>0, and Ω is a bounded domain of Rⁿ, N≥3. The functions ωᵢ, 1≤ i≤3, are nonnegative, continuous weights in Ω. We prove that there exists a region D in the λβ-plane where the Dirichlet problem has at least one positive solution. The novelty in this paper is that our result is valid for nonlinearities with growth higher than p in the gradient variable.