• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.1-10
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• 2012

Let ${\Omega}$ be a bounded subset of $\mathbb{R}^n$ with smooth boundary. We investigate the solvability for a class of the system of the nonlinear elliptic equations with Dirichlet boundary condition. Using the mountain pass theorem we prove that the system has at least one nontrivial solution.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.67-76
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• 2009

We show the existence of at least two nontrivial solutions for a class of the systems of the nonlinear elliptic equations with Dirichlet boundary condition under some conditions for the nonlinear term. We obtain this result by using the variational linking theory in the critical point theory.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.139-146
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• 2008

We investigate the uniqueness and multiplicity of solutions for the nonlinear elliptic system with Dirichlet boundary condition $$\{-{\Delta}u+g_1(u,v)=f_1(x){\text{ in }}{\Omega},\\-{\Delta}v+g_2(u,v)=f_2(x){\text{ in }}{\Omega},$$ where ${\Omega}$ is a bounded set in $R^n$ with smooth boundary ${\partial}{\Omega}$. Here $g_1$, $g_2$ are nonlinear functions of u, v and $f_1$, $f_2$ are source terms.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.895-909
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• 2017

This paper is concerned with the following fourth-order elliptic equations $${\Delta}^2u-{\Delta}u+V(x)u-{\frac{k}{2}}{\Delta}(u^2)u=f(x,u),\text{ in }{\mathbb{R}}^N$$, where $N{\leq}6$, ${\kappa}{\geq}0$. Under some appropriate assumptions on V(x) and f(x, u), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are extended.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.87-95
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• 1997

In general, the positive solution to biological reaction-diffusion equations is not unique. In this paper, we state the sufficient and necessary conditions of the existence of positive solutions, and give and the proof for the uniqueness of positive solutions for a certain elliptic interacting system.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.93-115
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• 2024

In this paper, the Gauge-Uzawa methods for the Darcy-Brinkman equations driven by temperature and salt concentration (DBTC) are proposed. The first order backward difference formula is adopted to approximate the time derivative term, and the linear term is treated implicitly, the nonlinear terms are treated semi-implicit. In each time step, the coupling elliptic problems of velocity, temperature and salt concentration are solved, and then the pressure is solved. The unconditional stability and error estimations of the first order semi-discrete scheme are derived, at the same time, the unconditional stability of the first order fully discrete scheme is obtained. Some numerical experiments verify the theoretical prediction and show the effectiveness of the proposed methods.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.77-90
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• 2004

The purpose of this paper is to give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system of the Dirichlet problem on the bounded domain ${\Omega}$ in $R^n$. 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. This result yields an algebraically computable criterion for the positive coexistence of competing species of animals in many biological models.

• Journal of Ocean Engineering and Technology
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• v.25 no.2
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• pp.62-66
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• 2011

In this paper, a liquid-filled long membrane container resting on a horizontal foundation is considered. All of the quantities are normalized to obtain similarity solutions. A system of nonlinear ordinary differential equations with undetermined boundary conditions is solved analytically. The integration of the curvature gives the solutions, which are expressed in terms of the elliptic integrals. A method for finding the shape and characteristic values is proposed for a given cross-sectional volume. The validity of these solutions is confirmed, and some results are shown for characteristic values and shapes.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.139-156
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• 2014

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order $h^{\frac{3}{2}}$ in the $L^2$-norm and order h in the $L^{\infty}$-norm for the control variable are proved.

• Journal of Theoretical and Applied Mechanics
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• v.2 no.1
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• pp.51-70
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• 1996

This study is generalization of the study of Miles[Physica 11D, 1984, pp.309-323]on the resonant motion of a spherical pendulum, which is equivalent to a particle on a spherical container subject to a linear, horizontal excitation. This study covers an arbitrary shape of container and a more general excitation (horizontal but elliptic motion). The averaging method is applied to reduce the governing equations to an autonomous system with cubic nonlinear terms, under the assumption of small amplitude of the container motion. It is shown that both the container shape and the excitation pattern affect the particle dynamics. Under the linear excitation, the anharmonic motion of the particle is possible only for a certain finite range of the parameter a controling the container shape. Stability of the particle's harmonic motion is also influenced by the excitation pattern; as the excitation trajectory becomes closer to a circle, the particle's motion has a stronger tendency to become stable and to follow the rotational direction of the excitation. Under a circular excitation, the motion is always stable and circular with the same rotational direction as the excitation. Analogy between the present model and that of the surface wave inside a circular is studied quantitatively.