Semilinear elliptic equations with subcritical absorption in Lipschitz domains

Let ΩΩ be a bounded Lipschitz domain in RNRN. Consider the equation (∗)−Δu+g(x,u)=0, g∈C(Ω×R)g∈C(Ω×R) and g(x,⋅)g(x,⋅) positive and increasing on R+R+, ∀x∈Ω∀x∈Ω. We say that gg is subcritical at y∈∂Ωy∈∂Ω if (∗)(∗) has a solution uk,yuk,y with boundary trace kδykδy, ∀k>0∀k>0. For a large family of functions gg, 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,yuk,y at yy. Some related results have been obtained in Marcus and Véron (2011) when g(t)=tqg(t)=tq. In the case that gg satisfies the Keller–Osserman condition we prove: if uu is a positive solution with strong singularity at y∈∂Ωy∈∂Ω (i.e., uu is not   dominated by a harmonic function) then u≥limk→∞uk,yu≥limk→∞uk,y. Finally, we extend estimates related to the Keller–Osserman condition, that are well-known in the case of C2C2 domains when gg is independent of the space variable, to Lipschitz domains and a large class of functions gg, including cases where g(x,t)→0g(x,t)→0 or g(x,t)→∞g(x,t)→∞as x→∂Ωx→∂Ω.