帶有Hatree和對數(shù)非線性項(xiàng)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(shù)非線性項(xiàng)和Hatree非線性項(xiàng)造成的影響，利用Ekeland變分方法，將方程轉(zhuǎn)化為求能量泛函的臨界點(diǎn)，然后利用Hatree非線性項(xiàng)的性質(zhì)和對對數(shù)非線性項(xiàng)的技巧性處理，證明了帶變號勢，對數(shù)非線性項(xiàng)和Hatree非線性項(xiàng)的Schrodinger問題的能量泛函滿足山路型結(jié)構(gòu)，利用序列的有界性得到了（PS）條件。結(jié)果表明，結(jié)合山路結(jié)構(gòu)，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(yù)期結(jié)果，對研究帶有雙變號勢的對數(shù)非線性項(xiàng)的Schrodinger方程解的存在性具有一定的借鑒意義。

關(guān)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(xiàn)標(biāo)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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