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Pré-Publication, Document De Travail Année : 2014

Elliptic equations involving general subcritical source nonlinearity and measures

Résumé

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm if}\ \alpha=1\qquad {\rm or\ \ in}\ \ \Omega^c \ \ {\rm if}\ \alpha\in(0,1),$$ where $\sigma,\varrho\ge0$, $\Omega$ is an open bounded $C^2$ domain in $\R^N$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ or Laplacian operator if $\alpha=1$, $\nu,\mu$ are suitable Radon measures and $g:\R_+\mapsto\R_+$ is a continuous function. We introduce an approach to obtain weak solutions for problem (E1)-(E2) when $g$ is integral subcritical and $\sigma,\varrho\ge0$ small enough.
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Dates et versions

hal-01072227 , version 1 (07-10-2014)

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  • HAL Id : hal-01072227 , version 1

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Laurent Veron, Huyuan Chen, Patricio Felmer. Elliptic equations involving general subcritical source nonlinearity and measures. 2014. ⟨hal-01072227⟩
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