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摘 要：為了深入阐述变号势对对数非线性项和Hatree非线性项造成的影响，利用Ekeland变分方法，将方程转化为求能量泛函的临界点，然后利用Hatree非线性项的性质和对对数非线性项的技巧性处理，证明了带变号势，对数非线性项和Hatree非线性项的Schrodinger问题的能量泛函满足山路型结构，利用序列的有界性得到了（PS）条件。结果表明，结合山路结构，能够获得问题非平凡解的存在性。研究方法在理论证明得到了良好的预期结果，对研究带有双变号势的对数非线性项的Schrodinger方程解的存在性具有一定的借鉴意义。
关键词：非线性泛函分析;Schrodinger方程;变号的势函数;对数不等式;变分方法;非平凡解
中图分类号：O175 文献标志码：A doi：10.7535/hbkd.2019yx06001
Abstract：In order to expound the influence of sign-changing potential on logarithmic nonlinearity and Hatree nonlinearity. By the variational method， a weak solution to the problem is a critical point of the energy functional. Then， by the logarithmic inequality， the energy functional of Schrodinger problem satisfies the mountain geometry and （PS） condition. The existence of nontrivial solutions is obtained by mountain pass theorem. The research method has good expected results in theoretical proof and laid a good foundation for the study of Schrodinger problem with logarithmic nonlinearity with double sign-changing potential.
Keywords：nonlinear functional analysis; Schrodinger equation; sign-changing potential; logarithmic inequality; variational method; nontrivial solution
参考文献/References：
[1] ELLIOTT H L. Existence and uniqueness of the minimizing solution of choquard＼"s nonlinear equation [J]. Studies in Applied Mathematics， 1977， 57（2）：93-105.
[2] LIONS P L. The Choquard equation and related questions [J]. Nonlinear analysis：Theory，Methods & Applications， 1980， 4（6）：1063-1072.
[3] MOROZ V， SCHAFTINGEN J V. Groundstates of nonlinear Choquard equations： Existence， qualitative properties and decay asympto-tics[J]. Journal of Functional Analysis， 2013， 265（2）：153-184.
[4] GAO Fashun， YANG Minbo. On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents[J]. Journal of Mathematical Analysis and Applications， 2017， 448（2）：1006-1041.
[5] DAVID R， SCHAFTINGEN J V. Odd symmetry of least energy nodal solutions for the Choquard equation [J]. Journal of Differential Equations， 2016：S0022039617305193.
[6] LI Guidong， LI Yongyong， TANG Chunlei， et al. Existence and concentrate behavior of ground state solutions for critical Choquard equation [J]. Applied Mathematics Letters，2019，81：96.
[7] MOROZ V， SCHAFTINGEN J V. A guide to the Choquard equation [J]. Journal of Fixed Point Theory and Applications， 2017，19（1）：773-813.
[8] TIAN Shuying. Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity [J]. Journal of Mathematical Analysis & Applications， 2017，454（2）：816-828. [9] SQUASSINA M， SZULKIN A. Multiple solutions to logarithmic Schrodinger equations with periodic potential [J]. Calculus of Variations and Partial Differential Equations， 2015， 54（1）：585-597.
[10] KAZUNAGA T， ZHANG Chengxiang. Multi-bump solutions for logarithmic Schrodinger equations [J]. Calculus of Variations and Partial Differential Equations， 2017， 2（2）：33-56.
[11] ARDILA A H， SQUASSINA M. Gausson dynamics for logarithmic Schrodinger equations [J]. Asymptotic Analysis， 2017， 107（3/4）：203-226.
[12] JI Chao， SZULKIN A. A logarithmic Schrodinger equation with asymptotic conditions on the potential[J]. Journal of Mathematical Analysis and Applcations，2016， 437（1）： 241-254.
[13] CARRILLO J A， NI Lei. Sharp logarithmic Sobolev inequalities on gradient solitons and applications [J]. Communications in Analysis & Geometry， 2009， 17（4）：721-753.
[14] JIA Wenyan， WANG Zuji. Multiple solution of p-Laplacian equation with the logarithmic nnlinearity[J]. Journal of North University of China， 2019，40（1）：26-33.
[15] ZHAO Li， HUANG Yongyan. The existence of the solution for Kirchhoff problem with sign-changing potential and logarithmic nonlinearity [J]. Journal of Shaanxi University of Science， 2019， 37（3）：176-184.
[16] WANG Jun， TIAN Lixin， XU Junxiang， et al. Erratum to： Existence and concentration of positive solutions for semilinear Schrodinger-Poisson systems in R3[J]. Calculus of Variations and Partial Differential Equations， 2013， 48（1/2）：275-276.
[17] LI Yuhua， LI Fuyi， SHI Junping. Existence and multiplicity of positive solutions to Schrodinger-Poisson type systems with critical nonlocal term [J]. Calculus of Variations & Partial Differential Equations， 2017，56（5）：134-151.
[18] WANG Zhengping， ZHOU Huansong. Sign-changing solutions for the nonlinear Schrodinger-Poisson system in R3[J]. Calculus of Variations & Partial Differential Equations， 2015， 52：927-943.
[19] LIU Hongliang， LIU Zhisu， XIAO Qizhen. Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity[J]. Applied Mathematics Letters， 2017，79：176-181.
[20] WILLEM M. Minimax theorems[J]. Progress in Nonlinear Differential Equations & Their Applications， 1996， 50（1）：139-141.
关键词：非线性泛函分析;Schrodinger方程;变号的势函数;对数不等式;变分方法;非平凡解
中图分类号：O175 文献标志码：A doi：10.7535/hbkd.2019yx06001
Abstract：In order to expound the influence of sign-changing potential on logarithmic nonlinearity and Hatree nonlinearity. By the variational method， a weak solution to the problem is a critical point of the energy functional. Then， by the logarithmic inequality， the energy functional of Schrodinger problem satisfies the mountain geometry and （PS） condition. The existence of nontrivial solutions is obtained by mountain pass theorem. The research method has good expected results in theoretical proof and laid a good foundation for the study of Schrodinger problem with logarithmic nonlinearity with double sign-changing potential.
Keywords：nonlinear functional analysis; Schrodinger equation; sign-changing potential; logarithmic inequality; variational method; nontrivial solution
参考文献/References：
[1] ELLIOTT H L. Existence and uniqueness of the minimizing solution of choquard＼"s nonlinear equation [J]. Studies in Applied Mathematics， 1977， 57（2）：93-105.
[2] LIONS P L. The Choquard equation and related questions [J]. Nonlinear analysis：Theory，Methods & Applications， 1980， 4（6）：1063-1072.
[3] MOROZ V， SCHAFTINGEN J V. Groundstates of nonlinear Choquard equations： Existence， qualitative properties and decay asympto-tics[J]. Journal of Functional Analysis， 2013， 265（2）：153-184.
[4] GAO Fashun， YANG Minbo. On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents[J]. Journal of Mathematical Analysis and Applications， 2017， 448（2）：1006-1041.
[5] DAVID R， SCHAFTINGEN J V. Odd symmetry of least energy nodal solutions for the Choquard equation [J]. Journal of Differential Equations， 2016：S0022039617305193.
[6] LI Guidong， LI Yongyong， TANG Chunlei， et al. Existence and concentrate behavior of ground state solutions for critical Choquard equation [J]. Applied Mathematics Letters，2019，81：96.
[7] MOROZ V， SCHAFTINGEN J V. A guide to the Choquard equation [J]. Journal of Fixed Point Theory and Applications， 2017，19（1）：773-813.
[8] TIAN Shuying. Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity [J]. Journal of Mathematical Analysis & Applications， 2017，454（2）：816-828. [9] SQUASSINA M， SZULKIN A. Multiple solutions to logarithmic Schrodinger equations with periodic potential [J]. Calculus of Variations and Partial Differential Equations， 2015， 54（1）：585-597.
[10] KAZUNAGA T， ZHANG Chengxiang. Multi-bump solutions for logarithmic Schrodinger equations [J]. Calculus of Variations and Partial Differential Equations， 2017， 2（2）：33-56.
[11] ARDILA A H， SQUASSINA M. Gausson dynamics for logarithmic Schrodinger equations [J]. Asymptotic Analysis， 2017， 107（3/4）：203-226.
[12] JI Chao， SZULKIN A. A logarithmic Schrodinger equation with asymptotic conditions on the potential[J]. Journal of Mathematical Analysis and Applcations，2016， 437（1）： 241-254.
[13] CARRILLO J A， NI Lei. Sharp logarithmic Sobolev inequalities on gradient solitons and applications [J]. Communications in Analysis & Geometry， 2009， 17（4）：721-753.
[14] JIA Wenyan， WANG Zuji. Multiple solution of p-Laplacian equation with the logarithmic nnlinearity[J]. Journal of North University of China， 2019，40（1）：26-33.
[15] ZHAO Li， HUANG Yongyan. The existence of the solution for Kirchhoff problem with sign-changing potential and logarithmic nonlinearity [J]. Journal of Shaanxi University of Science， 2019， 37（3）：176-184.
[16] WANG Jun， TIAN Lixin， XU Junxiang， et al. Erratum to： Existence and concentration of positive solutions for semilinear Schrodinger-Poisson systems in R3[J]. Calculus of Variations and Partial Differential Equations， 2013， 48（1/2）：275-276.
[17] LI Yuhua， LI Fuyi， SHI Junping. Existence and multiplicity of positive solutions to Schrodinger-Poisson type systems with critical nonlocal term [J]. Calculus of Variations & Partial Differential Equations， 2017，56（5）：134-151.
[18] WANG Zhengping， ZHOU Huansong. Sign-changing solutions for the nonlinear Schrodinger-Poisson system in R3[J]. Calculus of Variations & Partial Differential Equations， 2015， 52：927-943.
[19] LIU Hongliang， LIU Zhisu， XIAO Qizhen. Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity[J]. Applied Mathematics Letters， 2017，79：176-181.
[20] WILLEM M. Minimax theorems[J]. Progress in Nonlinear Differential Equations & Their Applications， 1996， 50（1）：139-141.