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Nonlinear elliptic equations with measure valued absorption potential

Abstract : We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary condition in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| ^{q−1} r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involving σ. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .
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  • HAL Id : hal-01726015, version 1
  • ARXIV : 1803.03150

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Nicolas Saintier, Laurent Veron. Nonlinear elliptic equations with measure valued absorption potential. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore In press. ⟨hal-01726015⟩

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