Positive symmetric solutions of singular semipositone boundary value problems
Using the method of upper and lower solutions, we prove that the singular boundary value problem, \[ -u'' = f(u) ~ u^{-\alpha} \quad \textrm{in} \quad (0, 1), \quad u'(0) = 0 = u(1) \, , \] has a positive solution when $0 < \alpha < 1$ and $f : \mathbb{R} \to \mathbb{R}$ is an...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2009-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=426 |
| _version_ | 1797830729029124096 |
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| author | M. Rudd Christopher Tisdell |
| author_facet | M. Rudd Christopher Tisdell |
| author_sort | M. Rudd |
| collection | DOAJ |
| description | Using the method of upper and lower solutions, we prove that the singular boundary value problem,
\[
-u'' = f(u) ~ u^{-\alpha} \quad \textrm{in} \quad (0, 1), \quad u'(0) = 0 = u(1) \, ,
\]
has a positive solution when $0 < \alpha < 1$ and $f : \mathbb{R} \to \mathbb{R}$ is an appropriate nonlinearity that is bounded below; in particular, we allow $f$ to satisfy the semipositone condition $f(0) < 0$. The main difficulty of this approach is obtaining a positive subsolution, which we accomplish by piecing together solutions of two auxiliary problems. Interestingly, one of these auxiliary problems relies on a novel fixed-point formulation that allows a direct application of Schauder's fixed-point theorem. |
| first_indexed | 2024-04-09T13:41:50Z |
| format | Article |
| id | doaj.art-9cc8cf6e4dc840c18dee593dc623b318 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:50Z |
| publishDate | 2009-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-9cc8cf6e4dc840c18dee593dc623b3182023-05-09T07:52:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752009-10-0120092411010.14232/ejqtde.2009.4.24426Positive symmetric solutions of singular semipositone boundary value problemsM. Rudd0Christopher Tisdell1University of Idaho, Moscow, ID, U.S.A.University of New South Wales, Sydney, AustraliaUsing the method of upper and lower solutions, we prove that the singular boundary value problem, \[ -u'' = f(u) ~ u^{-\alpha} \quad \textrm{in} \quad (0, 1), \quad u'(0) = 0 = u(1) \, , \] has a positive solution when $0 < \alpha < 1$ and $f : \mathbb{R} \to \mathbb{R}$ is an appropriate nonlinearity that is bounded below; in particular, we allow $f$ to satisfy the semipositone condition $f(0) < 0$. The main difficulty of this approach is obtaining a positive subsolution, which we accomplish by piecing together solutions of two auxiliary problems. Interestingly, one of these auxiliary problems relies on a novel fixed-point formulation that allows a direct application of Schauder's fixed-point theorem.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=426 |
| spellingShingle | M. Rudd Christopher Tisdell Positive symmetric solutions of singular semipositone boundary value problems Electronic Journal of Qualitative Theory of Differential Equations |
| title | Positive symmetric solutions of singular semipositone boundary value problems |
| title_full | Positive symmetric solutions of singular semipositone boundary value problems |
| title_fullStr | Positive symmetric solutions of singular semipositone boundary value problems |
| title_full_unstemmed | Positive symmetric solutions of singular semipositone boundary value problems |
| title_short | Positive symmetric solutions of singular semipositone boundary value problems |
| title_sort | positive symmetric solutions of singular semipositone boundary value problems |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=426 |
| work_keys_str_mv | AT mrudd positivesymmetricsolutionsofsingularsemipositoneboundaryvalueproblems AT christophertisdell positivesymmetricsolutionsofsingularsemipositoneboundaryvalueproblems |