Semilinear elliptic equations with subcritical absorption in Lipschitz domains

Abstract

Let Ω be a bounded Lipschitz domain in RN. Consider the equation (∗)−Δu+g(x,u)=0, g∈C(Ω×R) and g(x,⋅) positive and increasing on R+, ∀x∈Ω. We say that g is subcritical at y∈∂Ω if (∗) has a solution uk,y with boundary trace kδy, ∀k>0. For a large family of functions g, we establish existence and stability results for boundary value problems for (∗) with data given by measures concentrated on the set of subcritical points. In addition we describe the precise asymptotic behavior of uk,y at y. Some related results have been obtained in Marcus and Véron (2011) when g(t)=tq. In the case that g satisfies the Keller–Osserman condition we prove: if u is a positive solution with strong singularity at y∈∂Ω (i.e., u is not dominated by a harmonic function) then u≥limk→∞uk,y. Finally, we extend estimates related to the Keller–Osserman condition, that are well-known in the case of C2 domains when g is independent of the space variable, to Lipschitz domains and a large class of functions g, including cases where g(x,t)→0 or g(x,t)→∞as x→∂Ω.