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On a singularly perturbed equation with Neumann boundary condition. (English) Zbl 0898.35004
The paper deals with the study of the problem $$-\varepsilon^2\Delta u+u=u^q, \quad u>0\quad \text{in }\Omega, \qquad \partial u/\partial\nu=0\quad\text{on }\partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\bbfR^n$ and $q$ is a subcritical exponent. This problem may be viewed as a prototype of pattern formation in biology and it is also related to the steady state problem for a chemotactic aggregation model with logarithmic sensitivity. The above problem has been intensively studied by several authors (Lin, Ni, Takagi, Gui, Wang and others) who obtained strong results on the existence of a least energy solution, the study of asymptotics as $\varepsilon\to 0$ etc. The author applies a method which is originally due to Li and Nirenberg in order to construct solutions with single peak and multi-peak on $\partial\Omega$. Another result establishes the existence of multi-peak solutions with a prescribed number of local maximum points in $\overline\Omega$. The paper is interesting and the powerful methods developed by the authors allow the study of wide classes of boundary value problems.

35B25Singular perturbations (PDE)
35J65Nonlinear boundary value problems for linear elliptic equations
58C15Implicit function theorems and global Newton methods on manifolds
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