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Boundary trace of solutions of semilinear elliptic equalities and inequalities

Laurent Véron — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The boundary trace problem for positive solutions of - △ ⁢ u + g ⁢ x , u ≥ 0 is considered for nonlinearities of absorption type, and three different methods for defining the trace are compared. The boundary trace is obtained as a generalized Borel measure. The associated Dirichlet problem with boundary data in the set of such Borel measures is studied.

Boundary trace of positive solutions of nonlinear elliptic inequalities

Moshe Marcus;  Laurent Véron — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of - Δ u + g ( x , u ) ≥ 0 in a smooth domain Ω under very general assumptions on g . This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity...

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta;  Laurent Véron — 2013

Journal of the European Mathematical Society

For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on ∂ C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus;  Laurent Véron — 2004

Journal of the European Mathematical Society

Let Ω be a bounded domain of class C 2 in ℝ N and let K be a compact subset of ∂ Ω . Assume that q ≥ ( N + 1 ) / ( N − 1 ) and denote by U K the maximal solution of − Δ u + u q = 0 in Ω which vanishes on ∂ Ω ∖ K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ ∈ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

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