Elliptic equations with vertical asymptotes in the nonlinear term

Louis Dupaigne (1) , Augusto C. Ponce (2) , Alessio Porretta (3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 LJLL - Laboratoire Jacques-Louis Lions
3 DIPMAT - Dipartimento di Matematica [Roma II]

We study the existence of solutions of the nonlinear problem \begin{equation}\label{0.1} \left\{ \begin{alignedat}{2} - \Delta u + g(u) & = \mu && \quad \text{in } \Omega u & = 0 && \quad \text{on } \partial\Omega, \end{alignedat} \right. \end{equation} where $\mu$ is a bounded measure and $g$ is a continuous nondecreasing function such that $g(0) = 0$. In this paper, we assume that the nonlinearity $g$ satisfies \begin{equation}\label{0.2} \lim_{t \uparrow 1}{g(t)} = +\infty. \end{equation} Problem \eqref{0.1} need not have a solution for every measure $\mu$. We prove that, given $\mu$, there exists a "closest" measure $\mu^*$ for which \eqref{0.1} can be solved. We also explain how assumption \eqref{0.2} makes problem \eqref{0.1} different compared to the case where $g(t)$ is defined for every $t \in \RR$.

Mots clés: Elliptic equations involving measures; nonlinearities with vertical asymptotes; reduced measures

Domaines: Equations aux dérivées partielles [math.AP]

Louis Dupaigne, Augusto C. Ponce, Alessio Porretta. Elliptic equations with vertical asymptotes in the nonlinear term. Journal d'analyse mathématique, 2005, ?. ⟨hal-00015746⟩