Single blow-up solutions for a slightly subcritical biharmonic equation
<p>We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <...
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Hindawi Limited
2006-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/18387 |
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doaj-9890db29f3e84cb6be2e62bfd7706c882020-11-24T23:34:37ZengHindawi LimitedAbstract and Applied Analysis1085-33752006-01-012006Single blow-up solutions for a slightly subcritical biharmonic equation<p>We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>): <mml:math alttext="$Delta^2u=u^{9-varepsilon}$"> <mml:msup> <mml:mi>∆</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>9</mml:mn><mml:mo>−</mml:mo><mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> </mml:math>, <mml:math alttext="$u>0$"> <mml:mi>u</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math> in <mml:math alttext="$Omega$"> <mml:mi>Ω</mml:mi> </mml:math> and <mml:math alttext="$u=Delta u=0$"> <mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>∆</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn> </mml:math> on <mml:math alttext="$partialOmega$"> <mml:mo>∂</mml:mo><mml:mi>Ω</mml:mi> </mml:math>, where <mml:math alttext="$Omega$"> <mml:mi>Ω</mml:mi> </mml:math> is a smooth bounded domain in <mml:math alttext="$mathbb{R}^5$"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>5</mml:mn> </mml:msup> </mml:math>, <mml:math alttext="$varepsilon>0$"> <mml:mi>ε</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math>. We study the asymptotic behavior of solutions of (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>) which are minimizing for the Sobolev quotient as <mml:math alttext="$varepsilon$"> <mml:mi>ε</mml:mi> </mml:math> goes to zero. We show that such solutions concentrate around a point <mml:math alttext="$x_0 in Omega$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo><mml:mi>Ω</mml:mi> </mml:math> as <mml:math alttext="$varepsilon ightarrow 0$"> <mml:mi>ε</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn> </mml:math>, moreover <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> of the Robin's function, there exist solutions of (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>) concentrating around <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> as <mml:math alttext="$varepsilon ightarrow 0$"> <mml:mi>ε</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn> </mml:math>.</p>http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/18387 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| title |
Single blow-up solutions for a slightly subcritical biharmonic equation |
| spellingShingle |
Single blow-up solutions for a slightly subcritical biharmonic equation Abstract and Applied Analysis |
| title_short |
Single blow-up solutions for a slightly subcritical biharmonic equation |
| title_full |
Single blow-up solutions for a slightly subcritical biharmonic equation |
| title_fullStr |
Single blow-up solutions for a slightly subcritical biharmonic equation |
| title_full_unstemmed |
Single blow-up solutions for a slightly subcritical biharmonic equation |
| title_sort |
single blow-up solutions for a slightly subcritical biharmonic equation |
| publisher |
Hindawi Limited |
| series |
Abstract and Applied Analysis |
| issn |
1085-3375 |
| publishDate |
2006-01-01 |
| description |
<p>We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>): <mml:math alttext="$Delta^2u=u^{9-varepsilon}$"> <mml:msup> <mml:mi>∆</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mn>9</mml:mn><mml:mo>−</mml:mo><mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> </mml:math>, <mml:math alttext="$u>0$"> <mml:mi>u</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math> in <mml:math alttext="$Omega$"> <mml:mi>Ω</mml:mi> </mml:math> and <mml:math alttext="$u=Delta u=0$"> <mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>∆</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn> </mml:math> on <mml:math alttext="$partialOmega$"> <mml:mo>∂</mml:mo><mml:mi>Ω</mml:mi> </mml:math>, where <mml:math alttext="$Omega$"> <mml:mi>Ω</mml:mi> </mml:math> is a smooth bounded domain in <mml:math alttext="$mathbb{R}^5$"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>5</mml:mn> </mml:msup> </mml:math>, <mml:math alttext="$varepsilon>0$"> <mml:mi>ε</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn> </mml:math>. We study the asymptotic behavior of solutions of (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>) which are minimizing for the Sobolev quotient as <mml:math alttext="$varepsilon$"> <mml:mi>ε</mml:mi> </mml:math> goes to zero. We show that such solutions concentrate around a point <mml:math alttext="$x_0 in Omega$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo><mml:mi>Ω</mml:mi> </mml:math> as <mml:math alttext="$varepsilon ightarrow 0$"> <mml:mi>ε</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn> </mml:math>, moreover <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> of the Robin's function, there exist solutions of (<mml:math alttext="$P_varepsilon$"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> </mml:mrow> </mml:math>) concentrating around <mml:math alttext="$x_0$"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> as <mml:math alttext="$varepsilon ightarrow 0$"> <mml:mi>ε</mml:mi><mml:mo>→</mml:mo><mml:mn>0</mml:mn> </mml:math>.</p> |
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1725528516109271040 |