Concentration of solutions for some singularly perturbed mixed problems: asymptotics of minimal energy solutions
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 \textit{W. M. Ni} and \textit{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)].
scientific article; zbMATH DE number 5667945
scientific article