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<\v...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2021-09-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8651 |
| _version_ | 1797830441729785856 |
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| author | Shiyu Li Gongming Wei Xueliang Duan |
| author_facet | Shiyu Li Gongming Wei Xueliang Duan |
| author_sort | Shiyu Li |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-09T13:37:19Z |
| format | Article |
| id | doaj.art-700092911f124fcca40ddd56a938c0a8 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:19Z |
| publishDate | 2021-09-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-700092911f124fcca40ddd56a938c0a82023-05-09T07:53:11ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752021-09-0120216412510.14232/ejqtde.2021.1.648651On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficientsShiyu Li0Gongming Wei1Xueliang Duan2College of Science, University of Shanghai for Science and Technology, Shanghai, ChinaCollege of Science, University of Shanghai for Science and Technology, Shanghai, ChinaSchool of Mathematical Sciences, Zhejiang University, Hangzhou, ChinaIn 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.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8651asymptotic behaviorcritical sobolev exponenthardy exponentssingular coefficient |
| spellingShingle | Shiyu Li Gongming Wei Xueliang Duan On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients Electronic Journal of Qualitative Theory of Differential Equations 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 |
| title_full | On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients |
| title_fullStr | On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients |
| title_full_unstemmed | On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients |
| title_short | On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients |
| title_sort | on existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent and singular coefficients |
| topic | asymptotic behavior critical sobolev exponent hardy exponents singular coefficient |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8651 |
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