On some qualitative aspects for doubly nonlocal equations

In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation (-Delta)(s)u + mu u = (I-alpha * F(u))F'(u) in R-N (P) where N >= 2, s is an element of (0, 1), alpha is an element of (0, N), mu > 0 is fixed, (-Delta)(s) denotes the fractional Laplacian and I-alpha is the Riesz potential. Here F is an element of C-1(R) stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of (P). Then, by assuming F odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation (P). In particular, we extend some results contained in [23]. Similar qualitative properties of the ground states are obtained in the limiting case s = 1, generalizing some results by Moroz and Van Schaftingen in [52] when F is odd.