On semilinear elliptic equations with diffuse measures

We consider semilinear equation of the form , where L is the operator corresponding to a transient symmetric regular Dirichlet form , is a diffuse measure with respect to the capacity associated with , 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 . We also show that the solution is unique if f is nonincreasing in u.