On the boundary spike layer solutions to a singularly perturbed Neumann problem
This paper deals with the Neumann problem \[ \varepsilon^2\Delta u-u+u^p=0,\quad u>0\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}=0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\subset\mathbb{R}^n\) is a smooth bounded domain, \(1<p<(n+2)/(n- 2)\) when \(n\geq 3\) and \(1<p<\infty\) when \(n=1,2\). Associated with (1) is the functional \[ v\to I(\varepsilon,\Omega,v)= {1\over 2}\int_\Omega(\varepsilon^2|\nabla v|^2+v^2)- {1\over p+1}\int_\Omega v^{p+1}. \] Solutions \(u_\varepsilon\) of (1) are called single boundary peaked if \(\lim_{\varepsilon\to 0} \varepsilon^{-n}I_\varepsilon(u_\varepsilon)= {1\over 2} I(w)\), with \(I_\varepsilon(u_\varepsilon)= I(\varepsilon,\Omega,u_\varepsilon)\), \(I(w)= I(1,\mathbb{R}^n,w)\), where \(w\) is the positive radial solution of \(\Delta w-w+ w^p=0\) in \(\mathbb{R}^n\), \(w(z)\to 0\) as \(|z|\to\infty\), \(w(0)=\max_{z\in\mathbb{R}^n} w(z)\). The author states the theorem: if \(u_\varepsilon\) is a family of single boundary peaked solutions of (1), then, as \(\varepsilon\to 0\), \(u_\varepsilon\) has only one local maximum point \(P_\varepsilon\) and \(P_\varepsilon\in\partial\Omega\). Moreover, the tangential derivative of the mean curvature of \(\partial\Omega\) at \(P_\varepsilon\) tends to zero. A converse of this theorem is also investigated. The particular case of the least-energy solutions of (1) was studied previously by \textit{W-M. Ni} and \textit{I. Takagi} [Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042), and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)]. The proof is based on a decomposition of \(u_\varepsilon\) of the form \(u_\varepsilon=\alpha_\varepsilon w_\varepsilon+v_\varepsilon\), where \(w_\varepsilon\) is the solution of \(\varepsilon^2\Delta u-u+ w^p((x-P_\varepsilon)/\varepsilon)=0\) in \(\Omega\) and \(\partial u/\partial\nu=0\) on \(\partial\Omega\), and on fine estimates for \(\alpha_\varepsilon\in\mathbb{R}^+\) and the error term \(v_\varepsilon\in H^1(\Omega)\).
scientific article; zbMATH DE number 1000132