| In this thesis,we mainly apply the variational method and elliptic equations theories to study the existence,symmetry and nonsymmetry of ground state solutions and sign-changing solutions for quasilinear Schrodinger system and Choquard system.The specific contents are as follows:For the quasilinear Schr(?)dinger system:(?)where N≥3,α>1,β>1,2<α+β<4N/N-2 and k is a real number.In Chapter 1,we study the system(1)when k is equal to-1.By using the minimization method under Nehari-Pohozaev constraint and the action of G-invariant O(2)finite group,under certain assumptions of functions A(x)and B(x),we obtain the existence of positive ground state solutions and non-radially symmetric sign-changing solutions for the system(1).On the one hand,our results fill the gap that the parameter range of nonlinear term in quasilinear Schr(?)dinger system includes α+β∈(2,4).On the other hand,the existence of ground state solutions for periodic quasilinear Schr(?)dinger system seems to be the first result of this kind of system.In Chapter 2,we study the system(1)when k is greater than 0.By changing of variables,Pohozaev manifold and G-invariant O(2)finite group action,under certain assumptions of functions A(x)and B(x),we obtain that there is a positive ground state solution and a non radially symmetric sign-changing solution for the system(1).To the best of our knowledge,there there are no results on the existence of ground state solutions and sign-changing solutions for the quasilinear Schr(?)dinger system when k is greater than 0.For the Choquard system:(?)where N+α/N<p,q<N+α/N-2,N≥3 and 2<p,q<+∞,N=1,2.α∈(0,N)and Ia is a Riesz potential.In Chapter 3,under certain assumptions of functions A(x)and B(x),we use the minimization method under the constraints of Nehari manifold and Nehari nodal set to prove that there is a positive ground state solution and a sign-changing solution for the system(2).As far as we know,most of the previous conclusions consider the existence of ground state solutions for linearly coupled Choquard system.We study and obtain the existence of sign-changing solutions of nonlinear coupled Choquard system for the first time.In Chapter 4,when functions A(x)and B(x)are equal to 1,we first obtain the existence of odd symmetric ground state solutions of system(2)by using the Nehari manifold with odd symmetry.Secondly,by analyzing the asymptotic behavior of Riesz potential energy,we prove the existence of odd symmetric sign-changing solutions for system(2)when a approaches 0 and N,respectively.Finally,under the minimization constraint of the action of orthogonal transformation group,we obtain the existence of non radially symmetric sign-changing solutions for the system(2).The difficulty here is to deal with the nonlinear coupled term of Choquard type.We need to use the semigroup property of Riesz potential and the asymptotic behavior of energy many times.In Chapter 5,under the condition that functions A(x)and B(x)are equal and satisfy the periodic potential,using the generalized Nehari manifold and G-invariant O(2)finite group action,we obtain that the system(2)has a ground state solution and a non radially symmetric sign-changing solution.Most of the known conclusions are about the strongly indefinite problem for nonlinear Schrodinger equation and Schrodinger system.We extend this kind of problem to the Choquard system with strongly indefinite structure.The novelty here is that we study and obtain the existence of sign-changing solutions for strongly indefinite Choquard system. |
