Asymptotic behavior of the radial solution with nonnegative Dirichlet data on the annulus (Q939424)

The author considers the variational problem \(S_{p}=\{\inf\int_{\Omega}| \nabla u| ^{2}\in H_{r},\;u-\overline{\varphi}\in H^{1}_{0}(\Omega), \gamma=\| u\| _{p+1}\}\), where \(p>1, \Omega=\{x\in \mathbb{R}^{N} | 0<a<| x| <b\}\) is a symmetric abbulus in \(\mathbb{R}^{N} (N\geq 2), \overline{\varphi}\) is the harmonic extension of \(\varphi=c_{1},| x| =a, \varphi=c_{2},| x| =b, c_{1}, c_{2}\) are nonnegative constants. The asymptotic behavior of \(u_{p}\) as \(p\rightarrow\infty\) are studied.