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Elliptic equations with vertical asymptotes in the nonlinear term

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Abstract

We study the existence of solutions of the nonlinear problem {fx349-1} where μ is a bounded measure andg is a continuous nondecreasing function such thatg(0)=0. In this paper, we assume that the nonlinearityg satisfies {fx349-2} Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there exists a “closest” measure μ* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case whereg(t) is defined for everyt ∈ ℝ.

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Authors and Affiliations

  1. Laboratoire Amiénois de Mathématique Fondamentale et Appliquée Faculté de Mathématiques et d'Informatique, Université Picardie Jules Verne, 33, rue Saint-Leu, 80039, Amiens Cedex 1, France

    Louis Dupaigne

  2. Laboratoire de Mathématiques et Physique Théorique Fédération Denis Poisson, Université François Rabelais, 37200, Tours, France

    Augusto C. Ponce

  3. Dipartmento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italy

    Alessio Porretta

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  1. Louis Dupaigne
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  2. Augusto C. Ponce
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  3. Alessio Porretta
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Correspondence to Louis Dupaigne.

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Dupaigne, L., Ponce, A.C. & Porretta, A. Elliptic equations with vertical asymptotes in the nonlinear term. J. Anal. Math. 98, 349–396 (2006). https://doi.org/10.1007/BF02790280

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  • DOI: https://doi.org/10.1007/BF02790280

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