An eigenvalue problem for nonlocal equations

In this paper we study the existence of a positive weak solution for some classes of nonlocal equations under the Dirichlet boundary condition and involving the regional fractional Laplacian operator. More precisely, exploiting direct variational methods, we prove a characterization theorem for the existence of one weak solution for a nonlocal elliptic problem where the nonlinear term is a suitable continuous function. Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterizations properties proved for differential problems involving different elliptic operators. With respect to the cases studied in literature the nonlocal one considered here presents some additional difficulties so that a careful analysis of the fractional spaces involved, as well as some nonlocal $L^q$-estimates, is necessary.