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.
Highlights
Let E be a locally compact separable metric space, m be a positive Radon measure on E such that supp [m] = E, and let (E, D(E)) be a regular transient symmetric Dirichlet form on L2(E; m)
E, i.e. a bounded signed Borel measure on E which charges no set of capacity zero
In the present paper we provide a proof of the existence of a solution u in the sense of duality to (1.1) for f satisfying (1.2) and (1.5) under the following additional assumption on E: if {un} ⊂ De(E) and supn≥1 E(un, un) < ∞, up to a subsequence, {un} converges m-a.e
Summary
Let E be a locally compact separable metric space, m be a positive Radon measure on E such that supp [m] = E, and let (E, D(E)) be a regular transient symmetric Dirichlet form on L2(E; m). We consider semilinear equations of the form. In (1.1), f : E × R → R is a Caratheodory function satisfying the so-called “sign condition”:. E, i.e. a bounded signed Borel measure on E which charges no set of capacity zero. As for L, we assume that it is the operator corresponding to E, i.e. the unique nonpositive self-adjoint operator on L2(E; m) such that.
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