Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \[-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.\] Here \(\Omega\) is an annulus in \(\mathbb{R}^{n}\), \(n\geq 3\), \(\sigma...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2015-01-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3502.pdf |
| Summary: | In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \[-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.\] Here \(\Omega\) is an annulus in \(\mathbb{R}^{n}\), \(n\geq 3\), \(\sigma \lt 1\) and \(q\) is a positive function in \(\mathcal{C}_{loc}^{\gamma }(\Omega )\), \(0\lt\gamma \lt 1\), satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory. |
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| ISSN: | 1232-9274 |