On semilinear elliptic equations with diffuse measures

TL;DR

This paper studies the existence and uniqueness of solutions to a class of semilinear elliptic equations involving diffuse measures and Dirichlet forms, extending understanding of such equations without growth restrictions on the nonlinearity.

Contribution

It establishes existence of solutions under mild conditions and proves uniqueness when the nonlinearity is nonincreasing in the solution variable.

Findings

Existence of solutions under mild assumptions on the Dirichlet form.

Uniqueness of solutions when the nonlinearity is nonincreasing.

No growth condition on the nonlinear term $f(x,u)$.

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 $u$.