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Multiplicity of multiple interior peak solutions for some singularly perturbed Neumann problems. (English) Zbl 0916.35037
From the authors’ introduction: The present paper is concerned with the singularly perturbed elliptic problem: $$\varepsilon^2\Delta u- u+ u^p= 0,\quad u>0\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,$$ where $\Omega$ is a bounded smooth domain in $\bbfR^N$, $\varepsilon>0$ is a constant, $1< p< N+2/N- 2$ for $N\ge 3$ and $1< p<\infty$ for $N= 2$, and $\nu(x)$ denotes the normal derivative at $x\in\partial\Omega$. This is known as the stationary equation of the Keller-Segel system in chemotaxis. It can also be seen as the limiting stationary equation of the so-called Gierer-Meinhardt system in biological pattern formation. In this paper, we obtain a multiplicity result of $K$ interior peak solutions by using a category theory. Actually, we also able to handle more general nonlinearities than the power $u^p$. (Given two closed sets $A\subset B$, we say the category of $A\subset B$ is $k$, denoted by $\text{Cat}(A,B)= k$, where $k$ is the smallest number such that $A$ may be covered by $k$ closed contractible sets in $B$. We call the category of $B$ the strictly positive integer $\text{Cat}(B, B)$).
Reviewer: A.D.Osborne (Keele)

35J65Nonlinear boundary value problems for linear elliptic equations
35B25Singular perturbations (PDE)
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