一类带对数的半线性椭圆方程的多解

摘 要：本文研究了有界区域上一类带对数非线性项的半线性椭圆方程多解的存在性。首先，在Nehari流形上利用变分法证明该方程存在一个能量为负的正基态解；其次，证明问题对应的能量泛函满足（PS）条件，再借助Clark定理得到该问题的一列解。该结论在一定程度上补充完善了近期相关结果。

关键词：对数非线性；变分法；Nehari流形；多解

中图分类号：O177.91 文献标志码：A 文章编号：1673-5072（2024）06-0595-05

近年来，因为对数在量子力学、量子光学和核物理等领域的相关性，具有对数非线性项的各类偏微分问题引起了学者们的广泛关注。其中，关于带对数非线性项的半线性椭圆问题或者带位势的对数Schrdinger问题解的存在性的研究不在少数。例如，文献 ［1］关注了如下半线性椭圆问题

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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