## 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 $$\mathbb{R}^N$$, $$\varepsilon>0$$ is a constant, $$1< p< N+2/N- 2$$ for $$N\geq 3$$ 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 $$\text{Cat}(B, B)$$).

### MSC:

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

category theory
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