BOUNDARY BLOW-UP SOLUTIONS TO FRACTIONAL ELLIPTIC EQUATIONS IN A MEASURE FRAMEWORK

Let alpha is an element of (0,1), Q be a bounded open domain in R-N (N >= 2) with C-2 boundary partial derivative Omega and omega be the Hausdorff measure on partial derivative Omega. We denote by partial derivative(alpha)omega/partial derivative(n) over bar (alpha) a measure <partial derivative(alpha)omega/partial derivative(n) over right arrow (alpha), f > = integral partial derivative Omega partial derivative(alpha)f(x)partial derivative(n) over right arrow (alpha)(x) dw(x), f is an element of C-1 ((Omega) over bar), where (n) over right arrow (x) is the unit outward normal vector at point x is an element of partial derivative Omega. In this paper, we prove that problem (-Delta)(alpha)u + g(u) = k partial derivative(alpha)omega/partial derivative(n) over right arrow (alpha) in (Omega) over bar, (1) u = 0 in (Omega) over bar (c) admits a unique weak solution u(k) under the hypotheses that k > 0, (-Delta)(alpha) denotes the fractional Laplacian with alpha is an element of (0,1) and g is a nondecreasing function satisfying extra conditions. We prove that the weak solution of (1) is a classical solution of (-Delta)(alpha)u + g(u) = 0 in Omega, u = 0 in R-N\Omega, lim(x is an element of Omega,x ->partial derivative Omega) u(x) = +infinity.