COMPLEX AND REAL VALUED SOLUTIONS FOR FRACTIONAL HELMHOLTZ EQUATION

In this paper, we are concerned with the limiting absorption principle for the fractional Helmholtz equation (0.1) ( − ∆ ) s u − λ u = f ( x , u ) , in R n , where n ≥3, 0 +∞ and n n + 1 < s < n 2 are two real parameters. By establishing the boundedness estimate for the resolvent of fractional Helmholtz operator, we obtain the nontrivial L q ( R n ) complex valued solutions for (0.1). By setting up a dual variational framework, we also obtain the real valued solutions for (0.1) via a non-vanishing principle.