Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem
Futoshi Takahashi · Osaka City University (Osaka City University) · 2005
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.