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 \(\text{Cat}(B, B)\)).


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


category theory
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