Asymptotic profile of a radially symmetric solution with transition layers for an unbalanced bistable equation
In this article, we consider the semilinear elliptic problem $$ -varepsilon^{2}Delta u=h(|x|)^2(u-a(|x|))(1-u^2) $$ in $B_1(0)$ with the Neumann boundary condition. The function $a$ is a $C^1$ function satisfying $|a(x)|< 1$ for $xin [0,1]$ and $a'(0)=0$. In particular we consider the...
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| Format: | Article |
| Language: | English |
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Texas State University
2006-01-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2006/05/abstr.html |
| Summary: | In this article, we consider the semilinear elliptic problem $$ -varepsilon^{2}Delta u=h(|x|)^2(u-a(|x|))(1-u^2) $$ in $B_1(0)$ with the Neumann boundary condition. The function $a$ is a $C^1$ function satisfying $|a(x)|< 1$ for $xin [0,1]$ and $a'(0)=0$. In particular we consider the case $a(r)=0$ on some interval $Isubset [0,1]$. The function $h$ is a positive $C^1$ function satisfying $h'(0)=0$. We investigate an asymptotic profile of the global minimizer corresponding to the energy functional as $varepsilono 0$. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function $h$. |
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| ISSN: | 1072-6691 |