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On a singularly perturbed Neumann problem with the critical exponent. (English) Zbl 0997.35021
Summary: We consider the following singularly perturbed Neumann problem with the critical exponent: $\begin{cases} -\varepsilon^2\Delta u+u=u^{\frac{n+2}{n-2}},\;u>0,& \text{in }\Omega\\ \frac{\partial u}{\partial\nu}=0 & \text{on }\partial\Omega\end{cases}$ and show that as $$\varepsilon$$ tends to zero, the peak of the solutions tends to the boundary of the domain provided that the energy of the solutions remains below a certain natural level. This phenomenon is very different from the subcritical case.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B25 Singular perturbations in context of PDEs 35B33 Critical exponents in context of PDEs 35J60 Nonlinear elliptic equations
##### Keywords:
maximum principle; Hopf lemma; Green function; peak of solutions
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