Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian

We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $\Omega$ with two different boundary conditions at the same time: the shape of the solution outside $\Omega$ and a weighted limit to the boundary. We allow the nonlinearity to be positive or negative and we look for solutions blowing up at the boundary. Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative nonlinearity and as subsolutions in the positive case.