An elliptic semilinear equation with source term involving boundary measures: the subcritical case

Abstract

Abstract. We study the boundary behaviour of the nonnegative solutions of the semilinear elliptic equation in a bounded regular domain of R N (N 2), ( u + u q = 0 in u = on @ where 1 < q < (N + 1 ) =(N ; 1) and is a Radon measure on @ . We give a priori estimates and existence results. They lie on the study of the superharmonic functions in some weighted Marcinkiewicz spaces.

Elliptic Equations Involving Measures

“…Actually the representation formula is equivalent to the fact that u is a very weak solution of Problem 2.31 (see [14] for a proof). We set 48) and call it the Green potential of λ, and…”

“…In particular, using (2.50) and (2.52) it is not difficult to prove the following result (see [14] for a more general set of estimates in the case of the Laplacian operator).…”

“…All the L p (U )-spaces (1 ≤ p ≤ ∞) are relative to the measure µ. We denote by L p + (U ) their positive cones, 14) and…”

An elliptic semilinear equation with source term and boundary measure data: The supercritical case

“…for any (x, y, z) ∈ Ω × Ω × ∂Ω, see [2]. Since |ρ(x) − ρ(y)| ≤ |x − y| we have max ρ(x)ρ(y), |x − y| 2 ≍ max {|x − y|, ρ(x), ρ(y)} 2 .…”

“…The problem (1.3) has been first studied by Bidaut-Véron and Vivier [2] in the subcritical case 1 < q < N +1 N −1 with Ω bounded. They proved that (1.3) admits a nonnegative solution provided σ(∂Ω) is small enough.…”