Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations
<p/> <p>We study positive <inline-formula><graphic file="1687-2770-2006-87483-i2.gif"/></inline-formula> solutions to classes of boundary value problems of the form <inline-formula><graphic file="1687-2770-2006-87483-i3.gif"/></inline-...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2006-01-01
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| Series: | Boundary Value Problems |
| Online Access: | http://www.boundaryvalueproblems.com/content/2006/87483 |
| _version_ | 1817993165360070656 |
|---|---|
| author | Shivaji R Oruganti Shobha |
| author_facet | Shivaji R Oruganti Shobha |
| author_sort | Shivaji R |
| collection | DOAJ |
| description | <p/> <p>We study positive <inline-formula><graphic file="1687-2770-2006-87483-i2.gif"/></inline-formula> solutions to classes of boundary value problems of the form <inline-formula><graphic file="1687-2770-2006-87483-i3.gif"/></inline-formula> in <inline-formula><graphic file="1687-2770-2006-87483-i4.gif"/></inline-formula> on <inline-formula><graphic file="1687-2770-2006-87483-i5.gif"/></inline-formula>, where <inline-formula><graphic file="1687-2770-2006-87483-i6.gif"/></inline-formula> denotes the <inline-formula><graphic file="1687-2770-2006-87483-i7.gif"/></inline-formula>-Laplacian operator defined by <inline-formula><graphic file="1687-2770-2006-87483-i8.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i9.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i10.gif"/></inline-formula> is a parameter, <inline-formula><graphic file="1687-2770-2006-87483-i11.gif"/></inline-formula> is a bounded domain in <inline-formula><graphic file="1687-2770-2006-87483-i12.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i13.gif"/></inline-formula> with <inline-formula><graphic file="1687-2770-2006-87483-i14.gif"/></inline-formula> of class <inline-formula><graphic file="1687-2770-2006-87483-i15.gif"/></inline-formula> and connected (if <inline-formula><graphic file="1687-2770-2006-87483-i16.gif"/></inline-formula>, we assume that <inline-formula><graphic file="1687-2770-2006-87483-i17.gif"/></inline-formula> is a bounded open interval), and <inline-formula><graphic file="1687-2770-2006-87483-i18.gif"/></inline-formula> for some <inline-formula><graphic file="1687-2770-2006-87483-i19.gif"/></inline-formula> (semipositone problems). In particular, we first study the case when <inline-formula><graphic file="1687-2770-2006-87483-i20.gif"/></inline-formula> where <inline-formula><graphic file="1687-2770-2006-87483-i21.gif"/></inline-formula> is a parameter and <inline-formula><graphic file="1687-2770-2006-87483-i22.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i23.gif"/></inline-formula> function such that <inline-formula><graphic file="1687-2770-2006-87483-i24.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i25.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i26.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i27.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i28.gif"/></inline-formula>. We establish positive constants <inline-formula><graphic file="1687-2770-2006-87483-i29.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i30.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i31.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i32.gif"/></inline-formula>. Next we study the case when <inline-formula><graphic file="1687-2770-2006-87483-i33.gif"/></inline-formula> (logistic equation with constant yield harvesting) where <inline-formula><graphic file="1687-2770-2006-87483-i34.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i35.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i36.gif"/></inline-formula> function that is allowed to be negative near the boundary of <inline-formula><graphic file="1687-2770-2006-87483-i37.gif"/></inline-formula>. Here <inline-formula><graphic file="1687-2770-2006-87483-i38.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i39.gif"/></inline-formula> function satisfying <inline-formula><graphic file="1687-2770-2006-87483-i40.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i41.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i42.gif"/></inline-formula>, and <inline-formula><graphic file="1687-2770-2006-87483-i43.gif"/></inline-formula>. We establish a positive constant <inline-formula><graphic file="1687-2770-2006-87483-i44.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i45.gif"/></inline-formula> Our proofs are based on subsuper solution techniques.</p> |
| first_indexed | 2024-04-14T01:36:22Z |
| format | Article |
| id | doaj.art-144c4d14c84f4a86a28e9a914d48b94b |
| institution | Directory Open Access Journal |
| issn | 1687-2762 1687-2770 |
| language | English |
| last_indexed | 2024-04-14T01:36:22Z |
| publishDate | 2006-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-144c4d14c84f4a86a28e9a914d48b94b2022-12-22T02:19:55ZengSpringerOpenBoundary Value Problems1687-27621687-27702006-01-012006187483Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equationsShivaji ROruganti Shobha<p/> <p>We study positive <inline-formula><graphic file="1687-2770-2006-87483-i2.gif"/></inline-formula> solutions to classes of boundary value problems of the form <inline-formula><graphic file="1687-2770-2006-87483-i3.gif"/></inline-formula> in <inline-formula><graphic file="1687-2770-2006-87483-i4.gif"/></inline-formula> on <inline-formula><graphic file="1687-2770-2006-87483-i5.gif"/></inline-formula>, where <inline-formula><graphic file="1687-2770-2006-87483-i6.gif"/></inline-formula> denotes the <inline-formula><graphic file="1687-2770-2006-87483-i7.gif"/></inline-formula>-Laplacian operator defined by <inline-formula><graphic file="1687-2770-2006-87483-i8.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i9.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i10.gif"/></inline-formula> is a parameter, <inline-formula><graphic file="1687-2770-2006-87483-i11.gif"/></inline-formula> is a bounded domain in <inline-formula><graphic file="1687-2770-2006-87483-i12.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i13.gif"/></inline-formula> with <inline-formula><graphic file="1687-2770-2006-87483-i14.gif"/></inline-formula> of class <inline-formula><graphic file="1687-2770-2006-87483-i15.gif"/></inline-formula> and connected (if <inline-formula><graphic file="1687-2770-2006-87483-i16.gif"/></inline-formula>, we assume that <inline-formula><graphic file="1687-2770-2006-87483-i17.gif"/></inline-formula> is a bounded open interval), and <inline-formula><graphic file="1687-2770-2006-87483-i18.gif"/></inline-formula> for some <inline-formula><graphic file="1687-2770-2006-87483-i19.gif"/></inline-formula> (semipositone problems). In particular, we first study the case when <inline-formula><graphic file="1687-2770-2006-87483-i20.gif"/></inline-formula> where <inline-formula><graphic file="1687-2770-2006-87483-i21.gif"/></inline-formula> is a parameter and <inline-formula><graphic file="1687-2770-2006-87483-i22.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i23.gif"/></inline-formula> function such that <inline-formula><graphic file="1687-2770-2006-87483-i24.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i25.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i26.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i27.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i28.gif"/></inline-formula>. We establish positive constants <inline-formula><graphic file="1687-2770-2006-87483-i29.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i30.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i31.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i32.gif"/></inline-formula>. Next we study the case when <inline-formula><graphic file="1687-2770-2006-87483-i33.gif"/></inline-formula> (logistic equation with constant yield harvesting) where <inline-formula><graphic file="1687-2770-2006-87483-i34.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i35.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i36.gif"/></inline-formula> function that is allowed to be negative near the boundary of <inline-formula><graphic file="1687-2770-2006-87483-i37.gif"/></inline-formula>. Here <inline-formula><graphic file="1687-2770-2006-87483-i38.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i39.gif"/></inline-formula> function satisfying <inline-formula><graphic file="1687-2770-2006-87483-i40.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i41.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i42.gif"/></inline-formula>, and <inline-formula><graphic file="1687-2770-2006-87483-i43.gif"/></inline-formula>. We establish a positive constant <inline-formula><graphic file="1687-2770-2006-87483-i44.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i45.gif"/></inline-formula> Our proofs are based on subsuper solution techniques.</p>http://www.boundaryvalueproblems.com/content/2006/87483 |
| spellingShingle | Shivaji R Oruganti Shobha Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations Boundary Value Problems |
| title | Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations |
| title_full | Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations |
| title_fullStr | Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations |
| title_full_unstemmed | Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations |
| title_short | Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations |
| title_sort | existence results for classes of inline formula graphic file 1687 2770 2006 87483 i1 gif inline formula laplacian semipositone equations |
| url | http://www.boundaryvalueproblems.com/content/2006/87483 |
| work_keys_str_mv | AT shivajir existenceresultsforclassesofinlineformulagraphicfile16872770200687483i1gifinlineformulalaplaciansemipositoneequations AT orugantishobha existenceresultsforclassesofinlineformulagraphicfile16872770200687483i1gifinlineformulalaplaciansemipositoneequations |