Existence of nontrivial weak solutions for a quasilinear Choquard equation

We are concerned with the following quasilinear Choquard equation: −Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫ t 0 f(s)ds, −Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫t0f(s)ds, where 1<p<∞, Δpu=∇⋅(|∇u|p−2∇u) is the p-Laplacian operator, the potential function V:RN→(0,∞) is continuous and F∈C1(R,R). Here, Iα:RN→R is the Riesz potential of order α∈(0,p). We study the existence of weak solutions for the problem above via the mountain pass theorem and the fountain theorem. Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f.J. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). J.-M. Kim was supported by BK21 PLUS SNU Mathematical Sciences Division and National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1D1A1B03930422). J.-H. Bae was supported by BK21 PLUS SKKU Creative Mathematical Science Division and Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(NRF-2017R1D1A1B03031104). K. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070225)