Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions
Let $Omegasubset mathbb{R}^2$ be a bounded domain with $C^2$ boundary. In this paper, we are interested in the problem $$displaylines{ -Delta u+u = h(x,u) e^{u^2}/|x|^eta,quad u>0 quad ext{in } Omega, cr frac{partial u}{partial u}= lambda psi u^q quad ext{on }partial Omega, }$$ where $0in...
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| Format: | Article |
| Language: | English |
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Texas State University
2009-03-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2009/43/abstr.html |
| _version_ | 1818938726529630208 |
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| author | Bhatia Sumit Kaur K. Sreenadh |
| author_facet | Bhatia Sumit Kaur K. Sreenadh |
| author_sort | Bhatia Sumit Kaur |
| collection | DOAJ |
| description | Let $Omegasubset mathbb{R}^2$ be a bounded domain with $C^2$ boundary. In this paper, we are interested in the problem $$displaylines{ -Delta u+u = h(x,u) e^{u^2}/|x|^eta,quad u>0 quad ext{in } Omega, cr frac{partial u}{partial u}= lambda psi u^q quad ext{on }partial Omega, }$$ where $0in partial Omega$, $etain [0,2)$, $lambda>0$, $qin [0,1)$ and $psige 0$ is a H"older continuous function on $overline{Omega}$. Here $h(x,u)$ is a $C^{1}(overline{Omega}imes mathbb{R})$ having superlinear growth at infinity. Using variational methods we show that there exists $0<Lambda <infty$ such that above problem admits at least two solutions in $H^1(Omega)$ if $lambdain (0,Lambda)$, no solution if $lambda>Lambda$ and at least one solution when $lambda = Lambda$. |
| first_indexed | 2024-12-20T06:12:26Z |
| format | Article |
| id | doaj.art-8e1d6bb3b8fc430c8c9e4d3cca622a47 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T06:12:26Z |
| publishDate | 2009-03-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-8e1d6bb3b8fc430c8c9e4d3cca622a472022-12-21T19:50:38ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912009-03-01200943,111Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensionsBhatia Sumit KaurK. SreenadhLet $Omegasubset mathbb{R}^2$ be a bounded domain with $C^2$ boundary. In this paper, we are interested in the problem $$displaylines{ -Delta u+u = h(x,u) e^{u^2}/|x|^eta,quad u>0 quad ext{in } Omega, cr frac{partial u}{partial u}= lambda psi u^q quad ext{on }partial Omega, }$$ where $0in partial Omega$, $etain [0,2)$, $lambda>0$, $qin [0,1)$ and $psige 0$ is a H"older continuous function on $overline{Omega}$. Here $h(x,u)$ is a $C^{1}(overline{Omega}imes mathbb{R})$ having superlinear growth at infinity. Using variational methods we show that there exists $0<Lambda <infty$ such that above problem admits at least two solutions in $H^1(Omega)$ if $lambdain (0,Lambda)$, no solution if $lambda>Lambda$ and at least one solution when $lambda = Lambda$.http://ejde.math.txstate.edu/Volumes/2009/43/abstr.htmlMultiplicitynonlinear Neumann boundary conditionLaplace equation |
| spellingShingle | Bhatia Sumit Kaur K. Sreenadh Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions Electronic Journal of Differential Equations Multiplicity nonlinear Neumann boundary condition Laplace equation |
| title | Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions |
| title_full | Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions |
| title_fullStr | Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions |
| title_full_unstemmed | Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions |
| title_short | Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions |
| title_sort | multiple positive solutions for a singular elliptic equation with neumann boundary condition in two dimensions |
| topic | Multiplicity nonlinear Neumann boundary condition Laplace equation |
| url | http://ejde.math.txstate.edu/Volumes/2009/43/abstr.html |
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