On semilinear elliptic equations with diffuse measures

  • Tomasz Klimsiak
  • Andrzej RozkoszEmail author
Open Access


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.


Semilinear elliptic equation Dirichlet operator Measure data 

Mathematics Subject Classification

Primary 35J61 Secondary 35R06 


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Authors and Affiliations

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

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