带有Hatree和对数非线性项的Schrodinger方程非平凡解的存在性

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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
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