Semilinear elliptic equations and systems with diffuse measures

Abstract

Abstract. We study the equation −∆u + g(x, u) = µ, where g(·, s) is finite outside sets of zero H 1 -capacity, ∀s ∈ R, and µ is a diffuse measure. As an application, we provide a positive answer to a question of Lucio Boccardo concerning existence of solutions of an elliptic system with absorption.

Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data

“…Our results on the existence and uniqueness of solutions of (1.1) generalize known results in the sense that we consider semilinear parabolic systems with measure data (semilinear elliptic systems with measure data are considered in [15,23]). We should also stress that our results are proved for systems with f satisfying quite general condition (1.3) for which the usual monotonicity methods do not apply and we only require f to satisfy mild integrability condition (1.4) analogous to the integrability condition considered for elliptic equations or systems in [2,15,23].…”

“…We should also stress that our results are proved for systems with f satisfying quite general condition (1.3) for which the usual monotonicity methods do not apply and we only require f to satisfy mild integrability condition (1.4) analogous to the integrability condition considered for elliptic equations or systems in [2,15,23]. We also allow f to depend on x.…”

On the nonexistence of Green's function and failure of the strong maximum principle

“…These assertions can be established using [25,Section 9] and are related to the failure of the Hopf boundary lemma.…”

The sub-supersolution method for weak solutions

“…In general, (1.1) need not have a solution if (1.5) fails (see [13]). However, Orsina-Ponce [13] were recently able to prove existence of solutions of (1.1) (and (5.2) below) for some nonlinearities f which need not satisfy (1.5).…”