Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity

Abstract

We study existence problem for semilinear equations with Borel measure data and operator generated by a symmetric Markov semigroup. We assume merely that the nonlinear part satisfies the so-called sign condition. Using the method of sub and supersolutions we show the existence of maximal measure for which there exists a solution to the problem (the so-called reduced measure introduced by H. Brezis, M. Marcus and A.C. Ponce).

Renormalized solutions of semilinear elliptic equations with general measure data

“…One easily concludes now that u − u n L 1 (D) → 0, which shows that G # (f, µ) = {µ}. In [9,Section 6.4] it is shown that there exists a continuous metric projection onto G(f )…”

“…The measure µ * ,f is called the reduced measure. This notion was introduced in 2005 by Brezis, Marcus and Ponce [4] and further generalized to non-local operators in [8,9].…”

“…This is because we already have a fairly satisfactory knowledge on the objects G(f ) and Π f involved in (1.4). Firstly, in [9,Corollary 7.3] (see also [8,Theorem 5.13]) it has been proven that…”