Reduced measures for semilinear elliptic equations involving Dirichlet operators

Abstract

We consider elliptic equations of the form (E) - Au= f(x, u) + μ, where A is a negative definite self-adjoint Dirichlet operator, f is a function which is continuous and nonincreasing with respect to u and μ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where A= Δ and show basic properties of solutions of (E). We also prove Kato’s type inequality. Finally, we characterize the set of good measures in case f(u) = - up for some p> 1.