LARGE S-HARMONIC FUNCTIONS AND BOUNDARY BLOW-UP SOLUTIONS FOR THE FRACTIONAL LAPLACIAN

摘要

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 of the form { (-A)~su = ±f(x,u) inΩ u= gtin R~n Ω Eu= hton ∂Ω when the nonlinearity f and the boundary data g, h are positive, but allowing the right-hand side to be both positive or negative and looking for solutions that blow up at the boundary. The operator E is a weighted limit to the boundary: for example, if Ω is the ball B, there exists a constant C(n, s) > 0 such that Eu(θ) = C(n,s) lim_(x→θ)(x∈B)u(x) dist(x, ∂B)~(1-s), for all θ ∈ ∂B. 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 right-hand side and as subsolutions in the positive case.