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 ( P ε ): ∆ 2 u= u 9−ε , u>0 in Ω and u=∆u=0 on ∂Ω , where Ω is a smooth bounded domain in ℝ 5 , ε>0 . We study the asymptotic behavior of solutions of ( P ε ) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x 0 ∈Ω as ε→0 , moreover x 0 is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point x 0 of the Robin's function, there exist solutions of ( P ε ) concentrating around x 0 as ε→0 .

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