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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:

ε 2 Δu-u+u p =0,u>0inΩ,u ν=0onΩ,

where Ω is a bounded smooth domain in N , ε>0 is a constant, 1<p<N+2/N-2 for N3 and 1<p< for N=2, and ν(x) denotes the normal derivative at xΩ. 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 AB, we say the category of AB is k, denoted by 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 Cat(B,B)).

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