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Concentration of solutions for some singularly perturbed mixed problems: asymptotics of minimal energy solutions. (English) Zbl 1194.35037
This paper deals with positive solutions to the problem
$-\varepsilon^2\Delta u+u=u^p \text{ in } \Omega, \quad 1<p<\frac{n+2}{n-2} \tag{1}$
together with the boundary conditions
$\dfrac{\partial u}{\partial \nu}=0 \text{ on } \partial_{\mathcal N}\Omega ;\quad u=0 \text{ on }\partial_{\mathcal D} \Omega \tag{2}$ where $$\varepsilon$$ is a small positive parameter, $$\Omega$$ is a smooth bounded subset of $$\mathbb R^n$$, $$\partial_{\mathcal N}\Omega$$ and $$\partial _{\mathcal D}\Omega$$ are two disjoint open subsets of the boundary $$\partial\Omega$$ such that $$\partial\Omega=\overline{\partial_{\mathcal N}\Omega} \cup\overline{\partial_{\mathcal D}\Omega}$$ and $$\mathcal I_{\Omega}=:\overline{\partial_{\mathcal N}\Omega} \cap\overline{\partial_{\mathcal D}\Omega}$$ is an imbedded hypersurface. Let $$H$$ denote the mean curvature of $$\partial\Omega.$$ Then, the main result states that, as $$\varepsilon$$ tends to zero, the least energy solution to (1)-(2) has a unique maximum point which converges to $$Q\in\overline {\partial_{\mathcal N}\Omega}$$ such that $$H(Q)=\underset\overline{{\partial_{\mathcal N}\Omega}}{\max}H.$$ The proof is based on arguments of W. M. Ni and I.Takagi, [Commun. Pure Appl. Math. 44, No. 7, 819–851 (1991; Zbl 0754.35042)], and the authors’ paper [Arch. Ration. Mech. Anal. 196, No. 3, 907–950 (2010; Zbl 1214.35024)].

##### MSC:
 35B25 Singular perturbations in context of PDEs 35B34 Resonance in context of PDEs 35J20 Variational methods for second-order elliptic equations 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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