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(x, u) satisfies the sign condition in u and some weak integrability condition (no growth condition on f(x, u) 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.
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This work was supported by Narodowe Centrum Nauki (Grant No. 2016/23/B/ST1/01543).
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Klimsiak, T., Rozkosz, A. On semilinear elliptic equations with diffuse measures. Nonlinear Differ. Equ. Appl. 25, 35 (2018). https://doi.org/10.1007/s00030-018-0526-6
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DOI: https://doi.org/10.1007/s00030-018-0526-6