Existence Of Solutions To Choquard Equations

In this thesis, we study the existence of solutions for two types of Choquard equations. One type is general Choquard equation with nonlinearity, the other is the Choquard equation with nonlocal operator (-? +id)1/2. This thesis is constituted three chapters.In chapter 1, we introduce the background of the problems and main results of the thesis.In chapter 2, we study the existence of solutions for Choquard equation:-?u +u= q (I?* |u|p)|u|q-2u + p (I? * |u|q)|u|p-2u in RN (0.0.4)where N > 3, ? ? ? (0,N),I?:RN? R is the Riesz potential, p, q > 0 satisfying that Moreover, we prove a Pohozaev identity for problem (0.0.4), which implies the non-existence result for the problem when (p, q) does not satisfy the condition (0.0.5).In chapter 3, we consider the Choquard equation with nonlocal operator (-?+id)1/2:(-? + id)1/2u=(I? *|u|p)|u|p-2u in RN, u ? H1/2(RN). (0.0.6)We show that there is a ground state solution to problem (0.0.6) if N+?/N < p <N+?/N-1 and no solution to problem (0.0.6) if 0<p?N+?/N+1 or p ?N+?/N-1.