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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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