*(English)*Zbl 0916.35037

From the authorsâ€™ introduction: The present paper is concerned with the singularly perturbed elliptic problem:

where ${\Omega}$ is a bounded smooth domain in ${\mathbb{R}}^{N}$, $\epsilon >0$ is a constant, $1<p<N+2/N-2$ for $N\ge 3$ and $1<p<\infty $ for $N=2$, and $\nu \left(x\right)$ 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 (PDE) |