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浙江大学学报（理学版）
浙江大学学报（理学版）  2024, Vol. 51 Issue (3): 282-291    DOI: 10.3785/j.issn.1008-9497.2024.03.005
数学与计算机科学     
一类退化椭圆方程解的存在性与爆破行为
刘航,田书英()
武汉理工大学 理学院 数学系，湖北 武汉 430070
Existence and blow-up behavior of solutions for a class of degenerate elliptic equations
Hang LIU,Shuying TIAN()
Department of Mathematics，School of Science，Wuhan University of Technology，Wuhan 430070，China
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摘要：

研究了一类具有位势函数的锥形退化椭圆方程。通过探究约束极小化问题，建立了方程基态解的存在性定理，并分析了其爆破行为。证明了当参数满足一定条件时，约束极小化问题至少存在1个可达元，但当参数不满足此条件时，不存在可达元。详细分析了当参数趋近于临界值时，可达元的爆破行为。
关键词： 退化椭圆方程;  约束极小化问题;  基态解    
Abstract:

A class of conical degenerate elliptic equations with potential functions is studied. The existence of the ground state solution for the equation is established by studying the constrained minimization problem. The blow-up behavior is analyzed as well. Specifically, we prove that there exists at least one minimizer for the constraint minimization problem under certain conditions, but the minimizer does not exist when these conditions are unsatisfied. The blow-up behavior of the minimizer is analyzed in detail when the parameter approaches the critical value.
Key words: degenerate elliptic equation    constraint minimization problem    ground state solution
收稿日期: 2022-11-16 出版日期: 2024-05-07
CLC:  O 176.3  
基金资助: 国家自然科学基金资助项目(12071364)
通讯作者: 田书英     E-mail: sytian@whut.edu.cn
作者简介: 刘航（1997—），ORCID：https：//orcid.org/0000-0002-2180-2104，男，硕士，主要从事偏微分方程研究.
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引用本文:

刘航,田书英. 一类退化椭圆方程解的存在性与爆破行为[J]. 浙江大学学报（理学版）, 2024, 51(3): 282-291.

Hang LIU,Shuying TIAN. Existence and blow-up behavior of solutions for a class of degenerate elliptic equations. Journal of Zhejiang University (Science Edition), 2024, 51(3): 282-291.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2024.03.005    或    https://www.zjujournals.com/sci/CN/Y2024/V51/I3/282

1 CHEN H， LIU X C， WEI Y W. Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities［J］. Calculus of Variations and Partial Differential Equations， 2012， 43（3/4）： 463-484. DOI：10.1007/s00526-011-0418-7
doi: 10.1007/s00526-011-0418-7
2 CHEN H， LIU X C， WEI Y W. Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents［J］. Journal of Differential Equations， 2012， 252（7）： 4200-4228. DOI：10.1016/j.jde.2011.12.009
doi: 10.1016/j.jde.2011.12.009
3 CHEN H， WEI Y W. Existence of the eigenvalues for the cone degenerate p-Laplacian［J］. Chinese Annals of Mathematics （Series B）， 2021， 42（2）： 217-236. DOI：10.1007/s11401-021-0254-9
doi: 10.1007/s11401-021-0254-9
4 CHEN H， WEI Y W. Multiple solutions for nonlinear cone degenerate elliptic equations［J］. Communications on Pure and Applied Analysis， 2021， 20（7/8）： 2505-2518. DOI：10.3934/cpaa.2020272
doi: 10.3934/cpaa.2020272
5 CHEN H， LIU G W. Global existence and nonexistence for semilinear parabolic equations with conical degeneration［J］. Journal of Pseudo-Differential Operators and Applications， 2012， 3（3）： 329-349. DOI：10.1007/s11868-012-0046-9
doi: 10.1007/s11868-012-0046-9
6 ALIMOHAMMADY M， KALLEJI M K. Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration［J］. Journal of Functional Analysis， 2013， 265（10）： 2331-2356. DOI：10.1016/j.jfa.2013.07.013
doi: 10.1016/j.jfa.2013.07.013
7 LUO Y B， XU R Z， YANG C. Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities［J］. Calculus of Variations and Partial Differential Equations， 2022， 61（6）： 210. DOI：10.1007/s00526-022-02316-2
doi: 10.1007/s00526-022-02316-2
8 CHEN H， WEI Y W， ZHOU B. Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds［J］. Mathematische Nachrichten， 2012， 285（11/12）： 1370-1384. DOI：10.1002/mana.201100088
doi: 10.1002/mana.201100088
9 GIDAS B， NI W M， NIRENBERG L. Symmetry of Positive Solutions of Nonlinear Elliptic Equations in R n ［M］. New York/London： Academic Press， 1981： 369-402. doi:10.1080/03605308108820196
doi: 10.1080/03605308108820196
10 KWONG M K. Uniqueness of positive solutions of ∆u - u + up = 0 in R n ［J］. Archive for Rational Mechanics and Analysis， 1989， 105（3）： 243-266. DOI：10.1007/BF00251502
doi: 10.1007/BF00251502
11 LI Y， NI W M. Radial symmetry of positive solutions of nonlinear elliptic equations in R n ［J］. Communications in Partial Differential Equations， 1993， 18（5/6）： 1043-1054. doi:10.1080/03605309308820960
doi: 10.1080/03605309308820960
12 KEVIN M， JAMES S. Uniqueness of positive radial solutions of ∆u + f（u） = 0 in R n ［J］. Archive for Rational Mechanics and Analysis， 1987， 99（2）： 115-145. DOI：10.1007/BF00275874
doi: 10.1007/BF00275874
13 STRAUSS W A. Existence of solitary waves in higher dimensions［J］. Communications in Mathematical Physics， 1977， 55（2）： 149-162. DOI：10.1007/BF01626517
doi: 10.1007/BF01626517
14 LI G B. Some properties of weak solutions of nonlinear scalar field equations［J］. Annales Fennici Mathematici， 1990， 15（1）： 27-36. DOI：10.5186/aasfm.1990.1521
doi: 10.5186/aasfm.1990.1521
15 PUCCI P， SERRIN J. The Maximum Principle［M］. Basel： Birkhäuser Verlag， 2007. doi:10.1007/978-3-7643-8145-5
doi: 10.1007/978-3-7643-8145-5
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[2] 金永阳. 一类退化椭圆方程的弱解的正则性估计[J]. 浙江大学学报（理学版）, 2000, 27(4): 372-376.