Perturbative techniques for the construction of spike-layers

Andrea Malchiodi, ,Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
2019 Discrete and Continuous Dynamical Systems. Series A  
In this paper we survey some results concerning the construction of spike-layers, namely solutions to singularly perturbed equations that exhibit a concentration behaviour. Their study is motivated by the analysis of pattern formation in biological systems such as the Keller-Segel or the Gierer-Meinhardt's. We describe some general perturbative variational strategy useful to study concentration at points, and also at spheres in radially symmetric situations. 2010 Mathematics Subject
more » ... on. 35B25, 35B36, 35J20, 35J60, 53A07. Key words and phrases. Singularly perturbed elliptic problems, finite-dimensional reduction, pattern formation. Andrea Malchiodi has been supported by the project Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore and by MIUR Bando PRIN 2015 2015KB9WPT 001 . He is a member of GNAMPA as part of INdAM.
doi:10.3934/dcds.2020055 fatcat:4pcpo7mdsbbljbjc5ktmnrggba