On semilinear elliptic equations with diffuse measures

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 $u$.

References in corpus (2)

• Nonlocal equations with measure data
• Reduced measures for semilinear elliptic equations involving Dirichlet operators

Cited by in corpus (1)

• Renormalized solutions of semilinear elliptic equations with general measure data