一類帶臨界指數的半線性橢圓系統第二個正解的存在性

摘 要：本文研究了一類帶臨界指數的半線性橢圓系統，克服了臨界指數項產生的困難。首先，給出Nehari流形的定義，且將Nehari流形分為三部分；其次，證明系統對應的能量泛函滿足（PS）c條件，從而獲得泛函的緊性條件；最后，在Nehari流形上利用變分法證明該系統第二個正解的存在。該結果完善了帶臨界指數的半線性橢圓系統第二個正解結果，并給出這類系統問題新的可解性條件。

關鍵詞：半線性橢圓系統；臨界指數；正解；Nehari流形方法；變分法

中圖分類號：O177.91"" 文獻標志碼：A""" 文章編號：1673-5072（2023）02-0131-10

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Existence of the Second Positive Solution for A Class of Semilinear Elliptic System with Critical Exponent

LIAO Jia-fengab，ZHOU Xiua

（a.School of Mathematics amp; Information，b.College of Mathematics Education，China West Normal University，Nanchong Sichuan 637009，China）

Abstract：This paper talks about a class of semilinear elliptic system with critical exponent and solves the difficulty brought by critical exponent.Firstly，the definition of Nehari manifold is given，which is then divided into three parts.Secondly，the compactness condition is obtained by proving that the corresponding energy function satisfies the （PS）c condition.Finally，the existence of second positive solution for this system is proved by employing the variational method on Nehari manifold.The result has improved the existence of the second positive solution for semilinear elliptic system with critical exponent.Moreover，a new solvable condition for this system is presented.

Keywords：semilinear elliptic system;critical exponent;positive solution;Nehari manifold method;variational method