一類帶對數(shù)的半線性橢圓方程的多解

摘 要：本文研究了有界區(qū)域上一類帶對數(shù)非線性項的半線性橢圓方程多解的存在性。首先，在Nehari流形上利用變分法證明該方程存在一個能量為負的正基態(tài)解；其次，證明問題對應(yīng)的能量泛函滿足（PS）條件，再借助Clark定理得到該問題的一列解。該結(jié)論在一定程度上補充完善了近期相關(guān)結(jié)果。

關(guān)鍵詞：對數(shù)非線性；變分法；Nehari流形；多解

中圖分類號：O177.91 文獻標志碼：A 文章編號：1673-5072（2024）06-0595-05

近年來，因為對數(shù)在量子力學(xué)、量子光學(xué)和核物理等領(lǐng)域的相關(guān)性，具有對數(shù)非線性項的各類偏微分問題引起了學(xué)者們的廣泛關(guān)注。其中，關(guān)于帶對數(shù)非線性項的半線性橢圓問題或者帶位勢的對數(shù)Schrdinger問題解的存在性的研究不在少數(shù)。例如，文獻 ［1］關(guān)注了如下半線性橢圓問題

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Multiple Solutions for A Class of Semilinear Elliptic Equationswith Logarithmic Nonlinearity

LI Yu-hana，LIAO Jia-fengab

（a.School of Mathematics & Information，b.College of Mathematics Education，

China West Normal University，Nanchong Sichuan 637009，China）

Abstract：This paper talks about the existence of multiple solutions for a class of semilinear elliptic equations with logarithmic nonlinearity on a bounded domain.Firstly，the variational method is employed to prove the existence of a positive ground state solution on Nehari manifold，which has negative energy.Secondly，it is proved that the corresponding energy functional satisfies the （PS） condition.Then，a sequence of solutions are obtained by the Clark’s Theorem.The conclusion has complemented and improved the recent relevant results to some extent.

Keywords：logarithmic nonlinearity；variational method；Nehari manifold；multiple solutions