Elliptic Obstacle Problems With Natural Growth on the Gradient and Singular Nonlinear Term

Abstract

Abstract Given a bounded, open set Ω in ℝN (N ≥ 3), ψ∈ W1,p(Ω) (p > N) such that υ̸+ ∈ H0 1(Ω) ∩ L∞(Ω) and a suitable strictly positive (see (1.4)) function a ∈ Lq(Ω) with q > N/2, we prove the existence of positive solution w ∈ H0 1(Ω) of some variational inequality with a singular nonlinearity whose typical model is where the set of test functions K1 consists of all functions υ ∈ H0 1(Ω) ∩ L∞(Ω) such that υ(x) ≥ ψ(x) a.e. x ∈ Ω and supp (υ - ψ+) ⊂⊂ Ω. Bigger classes of test functions are also studied. We also recover the case in which the variational inequality reduces to an equation.