Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent
In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent $$displaylines{ -Delta u = lambda u - alpha u^p+ u^{2^*-1}, quad u >0 , quad hbox{in } Omega,cr u=0, quad hbox{on } partialOmega. }$$ where $Omega subset m...
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| Format: | Article |
| Language: | English |
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Texas State University
2006-10-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2006/135/abstr.thml |
| _version_ | 1811322917189844992 |
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| author | Yuanji Cheng |
| author_facet | Yuanji Cheng |
| author_sort | Yuanji Cheng |
| collection | DOAJ |
| description | In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent $$displaylines{ -Delta u = lambda u - alpha u^p+ u^{2^*-1}, quad u >0 , quad hbox{in } Omega,cr u=0, quad hbox{on } partialOmega. }$$ where $Omega subset mathbb{R}^n$, $nge 3 $ is a bounded $C^2$-domain $lambda>lambda_1$, $1<p < 2^* -1= frac{n+2}{n-2} $ and $alpha >0$ is a bifurcation parameter. Brezis and Nirenberg [2] showed that a lower order (non-negative) perturbation can contribute to regain the compactness and whence yields existence of solutions. We study the equation with an indefinite perturbation and prove a bifurcation result of two solutions for this equation. |
| first_indexed | 2024-04-13T13:44:44Z |
| format | Article |
| id | doaj.art-696d436dd70f44498ffb49d2813b34fe |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-13T13:44:44Z |
| publishDate | 2006-10-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-696d436dd70f44498ffb49d2813b34fe2022-12-22T02:44:31ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912006-10-01200613518Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponentYuanji ChengIn this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent $$displaylines{ -Delta u = lambda u - alpha u^p+ u^{2^*-1}, quad u >0 , quad hbox{in } Omega,cr u=0, quad hbox{on } partialOmega. }$$ where $Omega subset mathbb{R}^n$, $nge 3 $ is a bounded $C^2$-domain $lambda>lambda_1$, $1<p < 2^* -1= frac{n+2}{n-2} $ and $alpha >0$ is a bifurcation parameter. Brezis and Nirenberg [2] showed that a lower order (non-negative) perturbation can contribute to regain the compactness and whence yields existence of solutions. We study the equation with an indefinite perturbation and prove a bifurcation result of two solutions for this equation.http://ejde.math.txstate.edu/Volumes/2006/135/abstr.thmlCritical Sobolev exponentpositive solutionsbifurcation. |
| spellingShingle | Yuanji Cheng Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent Electronic Journal of Differential Equations Critical Sobolev exponent positive solutions bifurcation. |
| title | Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent |
| title_full | Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent |
| title_fullStr | Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent |
| title_full_unstemmed | Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent |
| title_short | Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent |
| title_sort | bifurcation of positive solutions for a semilinear equation with critical sobolev exponent |
| topic | Critical Sobolev exponent positive solutions bifurcation. |
| url | http://ejde.math.txstate.edu/Volumes/2006/135/abstr.thml |
| work_keys_str_mv | AT yuanjicheng bifurcationofpositivesolutionsforasemilinearequationwithcriticalsobolevexponent |