On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients

In this paper we study the existence and asymptotic behavior of solutions of $$-\Delta u=\mu\frac{u}{|x|^{2}}+|x|^{\alpha}u^{p(\alpha)-1-\varepsilon},\qquad u>0 \ \text{in}\ B_{R}(0)$$ with Dirichlet boundary condition. Here, $-2<\alpha<0$, $p(\alpha)=\frac{2(N+\alpha)}{N-2}$, $0<\vareps...

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Published inElectronic journal of qualitative theory of differential equations Vol. 2021; no. 64; pp. 1 - 25
Main Authors Li, Shiyu, Wei, Gongming, Duan, Xueliang
Format Journal Article
LanguageEnglish
Published University of Szeged 01.01.2021
Subjects
asymptotic behavior
critical sobolev exponent
hardy exponents
singular coefficient
Online AccessGet full text
ISSN1417-3875
1417-3875
DOI10.14232/ejqtde.2021.1.64

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Abstract In this paper we study the existence and asymptotic behavior of solutions of $$-\Delta u=\mu\frac{u}{|x|^{2}}+|x|^{\alpha}u^{p(\alpha)-1-\varepsilon},\qquad u>0 \ \text{in}\ B_{R}(0)$$ with Dirichlet boundary condition. Here, $-2<\alpha<0$, $p(\alpha)=\frac{2(N+\alpha)}{N-2}$, $0<\varepsilon<p(\alpha)-1$ and $p(\alpha)-1-\varepsilon$ is a nearly critical exponent. We combine variational arguments with the moving plane method to prove the existence of a positive radial solution. Moreover, the asymptotic behaviour of the solutions, as $\varepsilon\to0$, is studied by using ODE techniques.
AbstractList In this paper we study the existence and asymptotic behavior of solutions of $$-\Delta u=\mu\frac{u}{|x|^{2}}+|x|^{\alpha}u^{p(\alpha)-1-\varepsilon},\qquad u>0 \ \text{in}\ B_{R}(0)$$ with Dirichlet boundary condition. Here, $-2<\alpha<0$, $p(\alpha)=\frac{2(N+\alpha)}{N-2}$, $0<\varepsilon<p(\alpha)-1$ and $p(\alpha)-1-\varepsilon$ is a nearly critical exponent. We combine variational arguments with the moving plane method to prove the existence of a positive radial solution. Moreover, the asymptotic behaviour of the solutions, as $\varepsilon\to0$, is studied by using ODE techniques.
Author Duan, Xueliang
Li, Shiyu
Wei, Gongming
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Cites_doi 10.1007/978-3-540-74013-1
10.1007/s10231-005-0150-z
10.1016/S0022-247X(02)00292-5
10.1007/bf02786656
10.1007/978-94-010-9877-9_37
10.1512/iumj.1982.31.31056
10.1002/1097-0312(200102)54:2<229::aid-cpa4>3.0.co;2-i
10.1016/0022-0396(88)90147-7
10.1002/cpa.3160420304
10.1006/jdeq.1997.3375
10.1016/s0893-9659(04)90082-1
10.1090/s0002-9947-00-02560-5
10.1016/0022-1236(92)90070-y
10.1002/cpa.3160360405
10.1215/s0012-7094-91-06325-8
10.1112/jlms/s2-48.1.137
10.1016/0022-0396(87)90156-2
10.1007/bf01221125
10.1016/j.jde.2017.04.018
10.1016/0022-1236(73)90051-7
10.1007/s00526-009-0295-5
10.1142/s0219199702000725
10.1016/0022-0396(90)90093-5
10.1007/bf00431720
10.1142/s0219199717500997
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ref4
ref3
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ref5
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  doi: 10.1007/978-3-540-74013-1
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  doi: 10.1112/jlms/s2-48.1.137
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  doi: 10.1016/0022-0396(87)90156-2
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  doi: 10.1007/bf01221125
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  doi: 10.1016/0022-1236(73)90051-7
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  doi: 10.1142/s0219199702000725
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  doi: 10.1016/0022-0396(90)90093-5
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SubjectTerms asymptotic behavior
critical sobolev exponent
hardy exponents
singular coefficient
Title On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients
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