Semilinear equations for non-local operators

Wagner Vanja, University of Zagreb

We study the semilinear problem on a bounded open set, where the nonlocal operator is a generalization of the fractional Laplacian, associated with the generator of the subordinate Brownian motion, where the Laplace exponent of the subordinator is a complete Bernstein function. In order to discuss the existence and uniqueness of weak solutions to this Dirichlet problem, one has to impose not only standard complement data, but also a certain boundary condition on the boundary of the domain via a suitable boundary trace operator. We give existence results for a wide class of nonlinearities and the representation of the solution via suitable Green, Poisson, and Martin potentials associated with the nonlocal operator. These solutions are harmonically dominated and may explode at the boundary, but they have a finite boundary trace - therefore, we call them moderate solutions. Based on joint work with Ivan Bio\v ci\' c and Zoran Vondra\v cek.

Area: CS39 - Fractional operators and anomalous diffusions (Luisa Beghin)

Keywords: Non-local operators, Semilinear equations

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