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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)].

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
Full Text: DOI
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