Positive solutions for classes of multiparameter elliptic semipositone problems
We study positive solutions to multiparameter boundary-value problems of the form egin{gather*} - Delta u =lambda g(u)+mu f(u)quad ext{in } Omega \ u =0 quad ext{on } partial Omega , end{gather*} where $lambda >0$, $mu >0$, $Omega subseteq R^{n}$; $ngeq 2$ is a smooth bounded domai...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2007-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2007/96/abstr.html |
| _version_ | 1818187485891526656 |
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| author | Sumalee Unsurangsie Ratnasingham Shivaji Alfonso Castro Scott Caldwell |
| author_facet | Sumalee Unsurangsie Ratnasingham Shivaji Alfonso Castro Scott Caldwell |
| author_sort | Sumalee Unsurangsie |
| collection | DOAJ |
| description | We study positive solutions to multiparameter boundary-value problems of the form egin{gather*} - Delta u =lambda g(u)+mu f(u)quad ext{in } Omega \ u =0 quad ext{on } partial Omega , end{gather*} where $lambda >0$, $mu >0$, $Omega subseteq R^{n}$; $ngeq 2$ is a smooth bounded domain with $partial Omega $ in class $C^{2}$ and $Delta $ is the Laplacian operator. In particular, we assume $g(0)>0$ and superlinear while $f(0)<0$, sublinear, and eventually strictly positive. For fixed $mu$, we establish existence and multiplicity for $lambda $ small, and nonexistence for $lambda $ large. Our proofs are based on variational methods, the Mountain Pass Lemma, and sub-super solutions. |
| first_indexed | 2024-12-11T23:11:47Z |
| format | Article |
| id | doaj.art-66f3b005b0ea4cd8b4cc4adeacfa17db |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-11T23:11:47Z |
| publishDate | 2007-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-66f3b005b0ea4cd8b4cc4adeacfa17db2022-12-22T00:46:41ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912007-06-01200796110Positive solutions for classes of multiparameter elliptic semipositone problemsSumalee UnsurangsieRatnasingham ShivajiAlfonso CastroScott CaldwellWe study positive solutions to multiparameter boundary-value problems of the form egin{gather*} - Delta u =lambda g(u)+mu f(u)quad ext{in } Omega \ u =0 quad ext{on } partial Omega , end{gather*} where $lambda >0$, $mu >0$, $Omega subseteq R^{n}$; $ngeq 2$ is a smooth bounded domain with $partial Omega $ in class $C^{2}$ and $Delta $ is the Laplacian operator. In particular, we assume $g(0)>0$ and superlinear while $f(0)<0$, sublinear, and eventually strictly positive. For fixed $mu$, we establish existence and multiplicity for $lambda $ small, and nonexistence for $lambda $ large. Our proofs are based on variational methods, the Mountain Pass Lemma, and sub-super solutions.http://ejde.math.txstate.edu/Volumes/2007/96/abstr.htmlPositive solutionsmultiparametersmountain pass lemmasub-super solutionssemipositone |
| spellingShingle | Sumalee Unsurangsie Ratnasingham Shivaji Alfonso Castro Scott Caldwell Positive solutions for classes of multiparameter elliptic semipositone problems Electronic Journal of Differential Equations Positive solutions multiparameters mountain pass lemma sub-super solutions semipositone |
| title | Positive solutions for classes of multiparameter elliptic semipositone problems |
| title_full | Positive solutions for classes of multiparameter elliptic semipositone problems |
| title_fullStr | Positive solutions for classes of multiparameter elliptic semipositone problems |
| title_full_unstemmed | Positive solutions for classes of multiparameter elliptic semipositone problems |
| title_short | Positive solutions for classes of multiparameter elliptic semipositone problems |
| title_sort | positive solutions for classes of multiparameter elliptic semipositone problems |
| topic | Positive solutions multiparameters mountain pass lemma sub-super solutions semipositone |
| url | http://ejde.math.txstate.edu/Volumes/2007/96/abstr.html |
| work_keys_str_mv | AT sumaleeunsurangsie positivesolutionsforclassesofmultiparameterellipticsemipositoneproblems AT ratnasinghamshivaji positivesolutionsforclassesofmultiparameterellipticsemipositoneproblems AT alfonsocastro positivesolutionsforclassesofmultiparameterellipticsemipositoneproblems AT scottcaldwell positivesolutionsforclassesofmultiparameterellipticsemipositoneproblems |