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On the boundary spike layer solutions to a singularly perturbed Neumann problem. (English) Zbl 0873.35007
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 W-M. Ni and 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 $$v_\varepsilon\in H^1(\Omega)$$.
Reviewer: D.Huet (Nancy)

MSC:
 35B25 Singular perturbations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
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