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<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

Single blow-up solutions for a slightly subcritical biharmonic equation