Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources

摘要

In this paper we deal with elliptic and parabolic quasilinear problems with a singular quadratic lower order term depending on the gradient, i.e., with the equations —div(M(x)Δu) + B|Δ_u|~2/u~θ = u~r + f in Ω, and u'-div(M(x,t)Δu)+B|Δ_u|~2/u~θ=u~r inΩ×(0,T),u(x,0)=u_0(x) in Ω, with Ω a bounded open set of R~N, T > 0, M a bounded measurable uniformly elliptic matrix, B > 0, 0 < θ < 1 and 0 < r < 2 — θ. We will prove existence result for solutions under various assumptions on f and the initial datum u_0. Note that the elliptic equation is strongly related with the Euler—Lagrange equation of some integral functionals.