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Pré-Publication, Document De Travail Année : 2014

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

Résumé

We give new criteria for the existence of weak solutions to equation with source term \begin{align*}-\Delta u = u^q ~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega\end{align*}where $q>1$, $\Omega$ is a either a bounded smooth domain or $\mathbb{R}_+^{N}$ and $\sigma\in \mathfrak{M}^+(\partial\Omega)$ is a nonnegative Radon measure on $\partial\Omega$. In particular, one of the criteria is expressed in terms of some Bessel capacities on $\partial\Omega$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed term. \begin{align*} -\Delta u = |u|^{q_1-1}u|\nabla u|^{q_2} ~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega \end{align*} where $q_1,q_2\geq 0, q_1+q_2>1, q_2<2$, $\sigma\in \mathfrak{M}(\partial\Omega)$ is a Radon measure on $\partial\Omega$.
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Dates et versions

hal-01096025 , version 1 (16-12-2014)
hal-01096025 , version 2 (18-12-2014)
hal-01096025 , version 3 (20-12-2014)
hal-01096025 , version 4 (23-07-2015)
hal-01096025 , version 5 (09-09-2015)

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Marie-Françoise Bidaut-Véron, Giang Hoang, Quoc-Hung Nguyen, Laurent Véron. An elliptic semilinear equation with source term and boundary measure data: the supercritical case. 2014. ⟨hal-01096025v1⟩
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