Large Solutions For a Class of Nonlinear Elliptic Equations With Gradient Terms

Abstract

Abstract In this paper we prove the existence, in a suitable sense, of a large solution (a solution that blows-up at the boundary) for a class of nonlinear elliptic equations whose model is -Δ p u + u + u|▽u| q = f in Ω, p > 1, p - 1 < q ≤ p and f(x) ∈ L 1 (Ω). The main tool in order to prove it relies on approximating the problem with a more regular one, prove local (i.e. independent from the behavior on the boundary) a priori estimates and local compactness for truncations in W 1,p (Ω). Such scheme of the proof is also applied in order to prove the existence of a solution for equations of the type -Δ p u + u + u|▽u| q = f in ℝ N , where f(x) ∈ L 1 loc (ℝ N ) without any condition at infinity. Moreover, the local summability of the solution depending on the local summability of the datum is studied.