• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.245-251
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• 2019

Assume that ${\Omega}$ is a bounded domain in ${\mathbb{R}}^n$ with $n{\geq}2$. We study positive solutions to the problem, ${\Delta}u=u^p$ in ${\Omega}$, $u(x){\rightarrow}{\infty}$ as $x{\rightarrow}{\partial}{\Omega}$, where p > 1. Such solutions are called boundary blow-up solutions of ${\Delta}u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if 1 < p < ${\frac{n}{n-2}}$. In particular, when n = 2, there exists a boundary blow-up solution to ${\Delta}u=u^p$ for all $p{\in}(1,{\infty})$. We also prove the uniqueness under the additional assumption that the domain satisfies the condition ${\partial}{\Omega}={\partial}{\bar{\Omega}}$.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.377-385
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• 2002

In this paper, we concentrate on the uniquencess of the positive solution for the general elliptic system $\Delta$u+u($g_1$(u)-$g_2$(v))=0 $\Delta$u+u($h_1$(u)-$h_2$(v))=0 in$R_{+}$ $\times$ $\Omega$, $u\mid\partial\Omega = u\mid\partial\Omega = 0$. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

• 제어로봇시스템학회:학술대회논문집
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• 2003.10a
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• pp.25-29
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• 2003

LQ actuator selection problem for multi-input system discussed in this paper is to determine optimal actuator out of many actuators and input sequences so as to minimize the quadratic control performance. The solution of this problem depends on initial values and has a combinatorial property, so it is extremely difficult to get an optimal solution. For this difficulty, we proposed the concept of comparability of actuators and showed the uniqueness of the solution[1] . Further, to get general optimal solution for LQ problem with actuator selection strategies, we derived the equivalent condition for the comparability of actuator in single-input system . In this paper we extend this result to the case of multi-input system. The derived sufficient condition is applicable in the case of positive semi-definite comparability matrices.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.223-230
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• 2001

In this parper, we considered the uniqueness of positive time-solution to equation ${\Box}_g$u(t,$\chi$) - $c_n$u(t,$\chi$) + $c_n$u(t,$\chi$)$^[\frac{n+3}{n-3}]$ = 0, where $c_n$ = $\frac{n-1}{4n}$ and ${\Box}_g$ is the d'Alembertian for a Lorentzian warped manifold M = {a,$\infty$] $\times_f$ N.