一類(lèi)帶臨界指數(shù)的半線性橢圓系統(tǒng)第二個(gè)正解的存在性

摘 要：本文研究了一類(lèi)帶臨界指數(shù)的半線性橢圓系統(tǒng)，克服了臨界指數(shù)項(xiàng)產(chǎn)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(tǒng)對(duì)應(yīng)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(tǒng)第二個(gè)正解的存在。該結(jié)果完善了帶臨界指數(shù)的半線性橢圓系統(tǒng)第二個(gè)正解結(jié)果，并給出這類(lèi)系統(tǒng)問(wèn)題新的可解性條件。

關(guān)鍵詞：半線性橢圓系統(tǒng)；臨界指數(shù)；正解；Nehari流形方法；變分法

中圖分類(lèi)號(hào)：O177.91"" 文獻(xiàn)標(biāo)志碼：A""" 文章編號(hào)：1673-5072（2023）02-0131-10

參考文獻(xiàn)：

［1］ BRZIS H，NIRENBERG L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents［J］.Communications on Pure and Applied Mathematics，1983，36（4）：437-477.

［2］ ALVES C O，DE D C，F(xiàn)ILHO M，et al.On systems of elliptic equations involving subcritical or critical Sobolev exponents［J］.Nonlinear Anal，2000，42（5）：771-787.

［3］ BROWN K J，WU T F.A semilinear elliptic systems involving nonlinear boundary condition and sign-changing weight function［J］.Journal of Mathematical Analysis and Applications，2008，337（2）：1326-1336.

［4］ CHU C M，TANG C L.Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents［J］.Nonlinear Analysis，2009，71（11）：5118-5130.

［5］ HAN P G.Multiple positive solutions of nonhomogeneous elliptic system involving critical Sobolev exponents［J］.Nonlinear Analysis Theory Methods amp; Applications，2006，64（4）：869-886.

［6］ HSU T S.Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities［J］.Nonlinear Analysis Theory Methods amp; Applications，2009，（71）：2688-2698.

［7］ HSU T S，LIN H L.Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities［J］.Proceedings of the Royal Society of Edinburgh，2009，139（6）：1163-1177.

［8］ SHEN Y，ZHANG J H.Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent［J］.Nonlinear Analysis，2011，74（4）：1019-1030.

［9］ WU T F.The Nehari manifold for a semilinear elliptic system involving sign-changing weight function［J］.Nonlinear Analysis，2008，68（6）：1733-1745.

［10］廖家鋒，朱麗君.一類(lèi)帶有臨界指數(shù)的Schrdinger-Poisson系統(tǒng)非平凡解的存在性［J］.西華師范大學(xué)學(xué)報(bào)（自然科學(xué)版），2022，43（2）：149-155.

［11］廖家鋒，朱麗君，陳清方.一類(lèi)帶Hardy-Sobolev臨界指數(shù)的Kirchhoff型問(wèn)題的兩個(gè)正解［J］.西華師范大學(xué)學(xué)報(bào)（自然科學(xué)版），2020，41（3）：268-274.

［12］萬(wàn)優(yōu)艷，謝俊.一類(lèi)帶臨界指數(shù)增長(zhǎng)的橢圓型方程組兩個(gè)正解的存在性［J］.數(shù)學(xué)物理學(xué)報(bào)，2022，42（1）：103-130.

［13］HSU T S.Multiplicity of positive solutions for semilinear elliptic systems［J］.Abstract and Applied Analysis，2013，2013：746380.

［14］LIN H L.Multiple positive solutions for semilinear elliptic systems［J］.Journal of Mathematical Analysis amp; Applications，2012，391（1）：107-118.

［15］BROWN K J，ZHANG Y P.The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function［J］.Journal of Differential Equations，2003，193（2）：481-499.

［16］WU T F.On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function［J］.Journal of Mathematical Analysis and Applications，2006，318（1）：253-270.

［17］CAO D M，ZHOU H S.Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN［J］.Proceedings of the Royal Society of Edinburgh，1996，26（2）：443-463.

［18］VAZQUEZ J.A strong maximum principle for some quasilinear elliptic equations［J］.Applied Mathematics amp; Optimization.1984，（12）：191-202.

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