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Concentration of solutions for some singularly perturbed mixed problems: existence results. (English) Zbl 1214.35024
Let $$\Omega$$ be a bounded open subset of $$\mathbb R^n$$ with smooth boundary. Let $$p\in \,]1,\frac{n+2}{n-2}[$$. Let $$\nu$$ be the outward unit normal to $$\partial\Omega$$. Let $$\partial_{\mathcal D}\Omega$$, $$\partial_{\mathcal N}\Omega$$ be disjoint open subsets of the boundary $$\partial\Omega$$ of $$\Omega$$ such that the union of the closures of $$\partial_{\mathcal D}\Omega$$ and of $$\partial_{\mathcal N}\Omega$$ equals $$\partial\Omega$$.
The authors show that under suitable assumptions on the geometry of $$\partial\Omega$$, if $$\overline{Q}$$ belongs to the intersection of the closures of $$\partial_{\mathcal D}\Omega$$ and of $$\partial_{\mathcal N}\Omega$$, then the mixed boundary value problem
$-\varepsilon^2\Delta u + u=u^p\quad\text{in }\Omega, \qquad u>0 \quad\text{on }\Omega, \qquad \frac{\partial u}{\partial \nu}=0\quad\text{on }\partial_{\mathcal N}\Omega, \qquad u=0\quad\text{on }\partial_{\mathcal D}\Omega,$ admits a family of solutions $$u_\varepsilon$$ which concentrate at $$\overline{Q}$$ as the parameter $$\varepsilon>0$$ shrinks to $$0$$.

##### MSC:
 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35B25 Singular perturbations in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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