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Boundary concentration phenomena for a singularly perturbed elliptic problem. (English) Zbl 1124.35305
Summary: We exhibit new concentration phenomena for the equation $$-\varepsilon^2\Delta u+u=u^p$$ in a smooth bounded domain $$\Omega\subseteq\mathbb R^2$$ and with Neumann boundary conditions. The exponent $$p$$ is greater than or equal to 2 and the parameter $$\varepsilon$$ is converging to 0. For a suitable sequence $$\varepsilon_n\to 0$$ we prove the existence of positive solutions $$u_n$$ concentrating at the whole boundary of $$\Omega$$ or at some component.

MSC:
 35B25 Singular perturbations in context of PDEs 35J60 Nonlinear elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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