Large solutions to elliptic equations involving fractional Laplacian

Abstract

The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form

{(-Delta)(alpha)u(x) + vertical bar u vertical bar(p-1)u(x) = f(x), x is an element of Omega,

u(x) = 0, x is an element of(Omega) over bar (c)

lim(x is an element of Omega, x ->partial derivative Omega) u(x) = +infinity, where p > 1, Omega is an open bounded C-2 domain of R-N, N >= 2, the operator (-Delta)(alpha) with alpha is an element of (0, 1) is the fractional Laplacian and f: Omega -> R is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like d(x)(-2 alpha/p-1), when 1 + 2 alpha < p < 1 - 2 alpha/tau(0)(alpha), for some tau(0)(alpha) is an element of (-1, 0), and has infinitely many solutions with boundary behavior like d(x)(tau o(alpha)), when max{1 - 2 alpha/tau(0) + tau(0)(alpha)+1/tau(0), 1} < p < 1 - 2 tau/tau(0). Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).