Existence And Asymptotic Behaviour Of Solutions For Several Kinds Of Elliptic Equations With Convolutional Terms

This dissertation studies several types of elliptic equations with convolution terms,and studies the existence and asymptotic behavior of solutions by using the variational methods,The full text is divided into six chapters,the main content is as follows:Chapter 1 introduces the background and research significance of the problem,and describes the current research status and latest progress at home and abroad,as well as the main work of this thesis.Chapter 2 provides some basic concepts,lemmas and related notations,which are needed in this thesis.In Chapter 3,we investigate two classes of elliptic equations with singular potential and convolution terms.Firstly,we consider-Δu+V(|x|)u=(Iα*F(u))f(u),(0-6)where N≥3,α∈(0,N)and Iα is the Riesz potential.When f satisfies some appropriate conditions,we obtain the existence of ground state solutions of equation(0-6)by virtue of the Nehari manifold,mountain pass theorem and energy estimation.Subsequently,we consider-Δu+V(|x|)u+λφuu=|u|4α/4-αu+β|u|q-2u+|u|4u,x∈R3,(0-7)where φu=1/4π∫R3 u2(y)/|x-y|dy and q∈(2+4α/4-α,6).When λ>0 is small enough,we establish the existence of nontrivial solutions to equation(0-7)by perturbation method.In Chapter 4,we study two types of critical Choquard equations.Firstly,we consider-Δu-μ/|x|2 u=(Iα*|u|2α*|u|2α*-2u,x∈RN\{0},(0-8)where N≥3,α∈(0,N),μ∈(0,(N-2)2/2)and Iα is the Riesz potential.By using the rearrangement inequality and refined Sobolev inequality with Morrey norms,we prove the existence of nonnegative and nonincreasing radial solutions of equation(0-8)and by applying the Kelvin transformations we establish the asymptotic behavior of solutions.Subsequently,we apply the results of equation(0-8)to consider the following problem (?) where N≥3,α∈(0,N),μ∈(0,(N-2)2/4)and Iα is the Riesz potential.We obtain the existence of nontrivial solutions to equation(0-9)via the Nehari manifold and mountain pass theorem.Then,we give an estimate of the solution by means of the Moser iteration method.Finally,the asymptotic behavior of the solution is obtained through ground state representation and the Kelvin transformation.In Chapter 5,we study a class of elliptic equations with the van der Waals potential:-Δu+(Iα*|u|2)u=(Iβ*|u|2)u,x∈RN,(0-10)where N≥3,α,β∈(0,N),Iα and Iβ are Riesz potential.Under some certain conditions of α,β,we show the existence of ground state solutions of equation(0-10)by using the Pohozaev constraint method.Chapter 6 gives the conclusion and outlook of this thesis."0 drawings,0 tables,112 references"...