Existence Of Global Solutions For A Nonlinear Choquard Equation With Two Types Of Critical Exponents

In this paper we are concerned with the following nonlinear Choquard equation -△u + V（x）u =（I_α*|u|^p）|u|p-2u +|u|^p-2u+|u|^q-2 x∈R^N,where N≥3,0<α<N,Iα is the Riesz potential,V（x）is a continuous function,p= （N+）/N,g = 2*=（2N）/（N-2） is the critical exponent in the sense of the Sobolev inequality.According to the regularity and the Pohozaev identity of the solution,the Pohozaev-Palais-Smale sequence under the mountain pass type,we obtain the existence of global solutions under some assumptions on V（x）.This paper is organized as follows:In section 1,we mainly introduce the research background and main results.In section 2,we give the necessary preliminary knowledge to prove the conclusion,focus on giving the detailed calculation process of mountain pass level b estimation.In section 3,we use the critical points of mountain pass level b to construct the Pohozaev + Palais-Smale sequence.Finally,in the last section,we prove the existence of global solutions of the non-linear Choquard equation （?）.