In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))F′(u)in RN (P) where N≥2 , s∈(0,1) , α∈(0,N) , μ>0 is fixed, (−Δ)s denotes the fractional Laplacian and Iα is the Riesz potential. Here F∈C1(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.
On some qualitative aspects for doubly nonlocal equations
CINGOLANI S
;GALLO M
2022-01-01
Abstract
In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))F′(u)in RN (P) where N≥2 , s∈(0,1) , α∈(0,N) , μ>0 is fixed, (−Δ)s denotes the fractional Laplacian and Iα is the Riesz potential. Here F∈C1(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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
