Single blow-up solutions for a slightly subcritical biharmonic equation
We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε): ∆2u=u9−ε, u>0 in Ω and u=∆u=0 on ∂Ω, where Ω is a smooth bounded domain in ℝ5, ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotien...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2006-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/AAA/2006/18387 |
| Summary: | We consider a biharmonic equation under the Navier boundary
condition and with a nearly critical exponent (Pε): ∆2u=u9−ε, u>0 in Ω and u=∆u=0 on ∂Ω, where Ω is a smooth bounded domain in
ℝ5, ε>0. We study the asymptotic behavior
of solutions of (Pε) which are minimizing for the
Sobolev quotient as ε goes to zero. We show that such
solutions concentrate around a point x0∈Ω as ε→0, moreover x0 is a critical point of the Robin's function. Conversely, we show that for any
nondegenerate critical point x0 of the Robin's function, there
exist solutions of (Pε) concentrating around x0 as ε→0. |
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| ISSN: | 1085-3375 1687-0409 |