• Title, Summary, Keyword: system of nonlinear elliptic equations

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SOLVABILITY FOR A CLASS OF THE SYSTEMS OF THE NONLINEAR ELLIPTIC EQUATIONS

  • Jung, Tack-Sun;Choi, Q-Heung
    • 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.

BOUNDARY VALUE PROBLEM FOR A CLASS OF THE SYSTEMS OF THE NONLINEAR ELLIPTIC EQUATIONS

  • Jung, Tacksun;Choi, Q-Heung
    • 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.

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UNIQUENESS AND MULTIPLICITY OF SOLUTIONS FOR THE NONLINEAR ELLIPTIC SYSTEM

  • Jung, Tacksun;Choi, Q-Heung
    • 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.

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INFINITELY MANY SOLUTIONS FOR A CLASS OF MODIFIED NONLINEAR FOURTH-ORDER ELLIPTIC EQUATIONS ON ℝN

  • Che, Guofeng;Chen, Haibo
    • 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.

THE EXISTENCE, NONEXISTENCE AND UNIQUENESS OF GLOBAL POSITIVE COEXISTENCE OF A NONLINEAR ELLIPTIC BIOLOGICAL INTERACTING MODEL

  • Kang, Joon Hyuk;Lee, Jungho;Oh, Yun Myung
    • 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.

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ERROR ESTIMATES OF RT1 MIXED METHODS FOR DISTRIBUTED OPTIMAL CONTROL PROBLEMS

  • Hou, Tianliang
    • 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.

UNIQUENESS OF POSITIVE SOLUTIONS FOR PREDATOR-PREY INTERACTING SYSTEMS WITH NONLINEAR DIFFUSION RATES

  • Ahn, Inkyung
    • 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.

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Analytic Analysis of Liquid-Filled Membrane Container Resting on Horizontal Foundation with Given Cross-Sectional Volume (수평 지반에 놓인 액체 저장용 막구조물 형상의 단면 체적에 따른 해석적 해)

  • Choi, Yoon-Rak
    • 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.

UTD Analysis of the Subreflector of an Offxet Dual Reflector Antenna Mounted on a Satellite (UTD를 이용한 위성 탑재용 옵셋 복반사판 안테나의 부반사판 해석)

  • 임규태;이상설
    • Journal of the Korean Institute of Telematics and Electronics A
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    • v.32A no.1
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    • pp.79-88
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    • 1995
  • A subreflector of an offset dual reflecor antenna system mounted on satelite is analyzed by Uniform Geometrical Theory of Diffraction(UTD). In order to get the total electric field at an observation point, the reflected field and the diffracted fields obtained by UTD are summed. The reflected point and the diffracted points which have to satisfy nonlinear equations are obtained by numerical methods. The numerical results show taht diffracted fields eliminate the discontinuity of reflected fields at the shadow boundary. To show the validity of our results, the field pattern of the symmetric hyperboloidal reflector is computed and compared with S. W. Lee et al.'s results. At various obsevation angles, radiation patterns of offset ellipsoidal subreflectors offseted by a circular corn and by an elliptic corn are obrained, repectivelly.

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A Numerical Analysis of Free Surface Wave around a ship (선체주위 자유수면파의 수치해석)

  • Choon-Bum Hong;Seung-Hee Lee
    • Journal of the Society of Naval Architects of Korea
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    • v.31 no.3
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    • pp.80-86
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    • 1994
  • A numerical method for simulations of inviscid incompressible flow fields around a ship advancing on the free surface is developed. A body fitted coordinate system, generated by numerically solving elliptic type partial differential equations is used to conform the ship and free surface configurations. Three dimensional Euler equations transformed to the non-staggered body fitted coordinate system are discretised by finite difference method. Time and spatial derivatives are discretised by forward and centered differencings, respectively, and artificial dissipations are added to discretised convection terms for improvements of numerical stability. At each time steps, free surface elevations are recomputed to satisfy nonlinear free surface conditions. Poisson equations for pressure field are solved iteratively and the velocity field for next time step is extrapolated. To verify the developed numerical method, flow fields around a Wigley model are simulated(Fn=0.250-0.408) and compared with experimental data to show good agreements.

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