Existence And Multiplicity Of Solutions For The Equation With Hartree Nonlinearity

The study of differential equations with the purpose of application,or with the back-ground of physics,mechanics,and other subject problems,is not only the most important content in traditional applied mathematics,but also an important part of contemporary mathematics.At present,the main subject of differential equations is nonlinear differential equations,especially nonlinear partial differential equations.Nonlinear partial differential equations are an important branch of modern mathematics.It has been widely concerned by many authors.Moreover,as the most fundamental equations in the partial differential equa-tions,Choquard equation in quantum mechanics,gravity,magnetics play an important role,and have also drawn extensive attention over the past decade.In this paper,we use moun-tain pass theorem,perturbation method,global compactness lemma and some constrained variational method to obtain the existence of solutions of such equations.The thesis consists of two sections.In Chapter 1,the following Choquard equation with the nonlinear term containing both general singularity and quasi-critical nonlinearity(?)is discussed,where ? is a bounded domain in RN with smooth boundary (?)?,??(0,3),I?is the Riesz potential defined for every x?R3\{0)by (?) f?C(R+,R+),F(t)=?0t g(s)ds,0<h?L2(?),g?C((0,?),R+)and,h and g satisfies(h)h E L2(?),h(x)>0 a.e.x??,(g)g?C(0,?),R+)is nondecreasing,?01 g(s)ds<? and there exists ??(0,1)such that(?),f satisfies quasicritical growth:(f1)(?),(f2)(?),(f3)2F(t)?f(t)t,t?0.When the parameter ? is small enough,we obtain two solutions,one of the solutions is the local minimum point of corresponding functional,and the other is the limit of the mountain pass type solution to the perturbation problem.In Chapter 2,the following Kirchhoff-type equation with Hartree nonlinearity-(a+b?R3|?u|2)?u+V(x)u=(I?*F(u))f(u)in R3,is studied,where a>0,b?0,??(0,3),Ia is the Riesz potential.The potential V E C1(R3,R+)satisfies the following assumptions:(V1)for all x?R3,V(x)?linm|x|??V(x)=V?,and the strict inequality holds on a positive measure subset,(V2)there exists k?(0,a(2+?)/4)such that 0?(?V(x),x)?k/|x|2 for all x?R3\{0},where(·,·)denotes the inner product of R3.And F?C1(R,R),f=F' satisfies the Berestycki-Lions type assumptions:(f4)there exists C>0 such that for every tR,|tf(t)|?C(|t|3+?/3+|t|3+?),(f5)limt?0F(t)/|t|3+?/3=0 and lim|t|??F(t)/|t|3+?=0,(f6)there exists t0?R{0} such that F(t0)?0.We obtain the existence of a ground state solution to this problem by using mountain pass theorem,global compactness lemma and some constrained variational method.