JournalsaihpcVol. 22, No. 6pp. 799–815

Semilinear equations with exponential nonlinearity and measure data

  • Augusto C. Ponce

    Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, boîte courier 187, 75252 Paris cedex 05, France, Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA

  • Daniele Bartolucci

    Dipartimento di Matematica, Università di Roma “Tre”, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy

  • Fabiana Leoni

    Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazza A. Moro 2, 00185 Roma, Italy

  • Luigi Orsina

    Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazza A. Moro 2, 00185 Roma, Italy
Semilinear equations with exponential nonlinearity and measure data cover
Download PDF

Abstract

We study the existence and non-existence of solutions of the problem

{−Δu+eu−1=μin Ω,u=0on ∂Ω,\left\{\begin{matrix} −\mathrm{\Delta }u + \mathrm{e}^{u}−1 = \mu & \mathrm{in}\:\Omega , \\ u = 0 & \mathrm{on}\:\partial \Omega , \\ \end{matrix}\right.

where Ω is a bounded domain in RN\mathbb{R}^{N}, N⩾3N⩾3, and μ is a Radon measure. We prove that if μ⩽4πHN−2\mu ⩽4\pi \mathscr{H}^{N−2}, then (0.1) has a unique solution. We also show that the constant 4π in this condition cannot be improved.

Résumé

Nous étudions l'existence et la non existence des solutions de l'équation

{−Δu+eu−1=μdans Ω,u=0sur ∂Ω,\left\{\begin{matrix} −\mathrm{\Delta }u + \mathrm{e}^{u}−1 = \mu & \mathrm{dans}\:\Omega , \\ u = 0 & \mathrm{sur}\:\partial \Omega , \\ \end{matrix}\right.

où Ω est un domaine borné dans RN\mathbb{R}^{N}, N⩾3N⩾3, et μ est une mesure de Radon. Nous démontrons que si μ vérifie μ⩽4πHN−2\mu ⩽4\pi \mathscr{H}^{N−2}, alors le problème (0.2) admet une unique solution. Nous montrons que la constante 4π dans cette condition ne peut pas être améliorée.

Cite this article

Augusto C. Ponce, Daniele Bartolucci, Fabiana Leoni, Luigi Orsina, Semilinear equations with exponential nonlinearity and measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, pp. 799–815

DOI 10.1016/J.ANIHPC.2004.12.003