郝劍偉 黃永艷

摘 要：為了深入闡述變號勢對對數(shù)非線性項和Hatree非線性項造成的影響，利用Ekeland變分方法，將方程轉化為求能量泛函的臨界點，然后利用Hatree非線性項的性質和對對數(shù)非線性項的技巧性處理，證明了帶變號勢，對數(shù)非線性項和Hatree非線性項的Schrodinger問題的能量泛函滿足山路型結構，利用序列的有界性得到了（PS）條件。結果表明，結合山路結構，能夠獲得問題非平凡解的存在性。研究方法在理論證明得到了良好的預期結果，對研究帶有雙變號勢的對數(shù)非線性項的Schrodinger方程解的存在性具有一定的借鑒意義。

關鍵詞：非線性泛函分析;Schrodinger方程;變號的勢函數(shù);對數(shù)不等式;變分方法;非平凡解

中圖分類號：O175 ? 文獻標志碼：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

參考文獻/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.