TY - JOUR

T1 - Locator function for concentration points in a spatially heterogeneous semilinear neumann problem

AU - Takagi, Izumi

AU - Yamamoto, Hiroko

N1 - Funding Information:
Acknowledgement. This work has been supported in part by JSPS KAK-ENHI (grant nos. 22244010 and 26610027).
Publisher Copyright:
Indiana University Mathematics Journal ©

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Anisotropic diffusion

KW - Concentration point

KW - Ground-state solution

KW - Locator function

KW - Pattern formation

KW - Semilinear elliptic equation

KW - Singular perturbation

KW - Spatial heterogeneity

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U2 - 10.1512/iumj.2019.68.7560

DO - 10.1512/iumj.2019.68.7560

M3 - Article

AN - SCOPUS:85064276034

VL - 68

SP - 63

EP - 103

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 1

ER -