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

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Abstract We study the existence problem for semilinear equations (E): −Au = f(⋅, u) + μ, with Borel measure μ and operator A that generates a symmetric Markov semigroup. We merely assume that the nonlinear part f is a Carathéodory function satisfying the so-called sign condition. We extend the method of sub and supersolutions for (E) and prove that if such exist, then there exists a solution to (E) (we do not even assume that the subsolution is less than or equal to the supersolution!). We further show that for any μ there exists a unique metric projection μ ̂ $\hat{\mu }$ of μ onto the set of good measures, i.e. Borel measures for which there exists a solution to (E).