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Zbl 0873.35007
Wei, Juncheng
On the boundary spike layer solutions to a singularly perturbed Neumann problem. (English)
[J] J. Differ. Equations 134, No.1, 104-133 (1997). ISSN 0022-0396

This paper deals with the Neumann problem $$\varepsilon^2\Delta u-u+u^p=0,\quad u>0\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}=0\quad\text{on }\partial\Omega,\tag1$$ where $\Omega\subset\bbfR^n$ is a smooth bounded domain, $1<p<(n+2)/(n- 2)$ when $n\ge 3$ 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 $I_\varepsilon(u_\varepsilon)= I(\varepsilon,\Omega,u_\varepsilon)$, $I(w)= I(1,\bbfR^n,w)$, where $w$ is the positive radial solution of $\Delta w-w+ w^p=0$ in $\bbfR^n$, $w(z)\to 0$ as $|z|\to\infty$, $w(0)=\max_{z\in\bbfR^n} w(z)$.\par 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 {\it W-M. Ni} and {\it 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)].\par 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 $\alpha_\varepsilon\in\bbfR^+$ and the error term $v_\varepsilon\in H^1(\Omega)$.
[D.Huet (Nancy)]

MSC 2000:: *35B25 Singular perturbations (PDE)
35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B40 Asymptotic behavior of solutions of PDE

Keywords: single peaked solutions; mean curvature

Citations: Zbl 0754.35042; Zbl 0796.35056

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