## Osaka Journal of Mathematics

### Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem

Futoshi Takahashi

#### Abstract

In this paper, we study the following fourth order elliptic problem $(E_p)$: \begin{eqnarray*} (E_p) \left \{ \begin{array}{l} \Delta^2 u = u^p \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right. \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and $p >1$ is any positive number.

We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$. Combining the arguments of Ren-Wei [8] and Wei [10], we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains, they have one or two peaks'' away form the boundary. If it happens that the only one peak point appears, we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$ under the Navier boundary condition.

#### Article information

Source
Osaka J. Math., Volume 42, Number 3 (2005), 633-651.

Dates
First available in Project Euclid: 21 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1153494506

Mathematical Reviews number (MathSciNet)
MR2166726

Zentralblatt MATH identifier
1165.35352

#### Citation

Takahashi, Futoshi. Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem. Osaka J. Math. 42 (2005), no. 3, 633--651. https://projecteuclid.org/euclid.ojm/1153494506