帶有Hatree和對數非線性項的Schrodinger方程非平凡解的存在性

郝劍偉 黃永艷

摘 要：為了深入闡述變號勢對對數非線性項和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非線性項的技巧性處理，證明了帶變號勢，對數非線性項和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;對數不等式;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

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