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On semilinear elliptic equations with diffuse measures

  • Tomasz Klimsiak
  • Andrzej Rozkosz
Open Access
Article

Abstract

We consider semilinear equation of the form \(-Lu=f(x,u)+\mu \), where L is the operator corresponding to a transient symmetric regular Dirichlet form \({\mathcal {E}}\), \(\mu \) is a diffuse measure with respect to the capacity associated with \({\mathcal {E}}\), and the lower-order perturbing term f(xu) satisfies the sign condition in u and some weak integrability condition (no growth condition on f(xu) as a function of u is imposed). We prove the existence of a solution under mild additional assumptions on \({\mathcal {E}}\). We also show that the solution is unique if f is nonincreasing in u.

Keywords

Semilinear elliptic equation Dirichlet operator Measure data 

Mathematics Subject Classification

Primary 35J61 Secondary 35R06 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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