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 |
| Published: |
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 |
| Summary: | 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. |
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| ISSN: | 1417-3875 |