ON NONLOCAL NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN

In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional alpha-Laplacian on R-n, 0 < alpha < n. We show that the problem has infinitely many positive solutions in C-tau (R-n) boolean AND H-loc(alpha/2) (R-n). Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on R-n. We also show for alpha is an element of (0, 2) that in some cases, by the use of Hardy's inequality, there is a nontrivial non-negative H-loc(alpha/2)(R-n) weak solution to the problem (-Delta)(alpha/2)u(x) = K(x)u(p) in R-n, where K(x) = K(vertical bar x vertical bar) is a non-negative non-increasing continuous radial function in R-n and p > 1.