Locator function for concentration points in a spatially heterogeneous semilinear neumann problem

Izumi Takagi, Hiroko Yamamoto

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

This paper is concerned with qualitative properties of solutions of the boundary value problem for a second-order semilinear elliptic equation with a small parameter in the coefficients of the highest-order differential operator. We study the asymptotic behavior of a family of solutions, called ground-state solutions, as the parameter approaches zero, in the case where all the coefficients depend on the spatial variable. We prove that a ground-state solution has only one local maximum, hence the global maximum, and it is achieved at exactly one point. Moreover, the distribution of the ground-state solution is concentrated in a very narrow region around this unique maximum point. To locate the concentration point, we introduce the locator function defined by using the coefficients of the equation, and prove, for instance, that if the global minimum of the locator function over the domain is strictly smaller than a half of the minimum over the boundary, then the concentration point is in the interior of the domain and is in a small neighborhood of the global minimum point of the locator function. This shows a sharp contrast with the case of constant coefficients, where ground-state solutions are concentrated at a boundary point.

Original languageEnglish
Pages (from-to)63-103
Number of pages41
JournalIndiana University Mathematics Journal
Volume68
Issue number1
DOIs
Publication statusPublished - 2019

Keywords

  • Anisotropic diffusion
  • Concentration point
  • Ground-state solution
  • Locator function
  • Pattern formation
  • Semilinear elliptic equation
  • Singular perturbation
  • Spatial heterogeneity

ASJC Scopus subject areas

  • Mathematics(all)

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