• Title/Summary/Keyword: Elliptic Equation

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Oscillation of Certain Second Order Damped Quasilinear Elliptic Equations via the Weighted Averages

  • Xia, Yong;Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • v.47 no.2
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    • pp.191-202
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    • 2007
  • By using the weighted averaging techniques, we establish oscillation criteria for the second order damped quasilinear elliptic differential equation $$\sum_{i,j=1}^{N}D_i(a_{ij}(x){\parallel}Dy{\parallel}^{p-2}D_jy)+{\langle}b(x),\;{\parallel}Dy{\parallel}^{p-2}Dy{\rangle}+c(x)f(y)=0,\;p>1$$. The obtained theorems include and improve some existing ones for the undamped halflinear partial differential equation and the semilinear elliptic equation.

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GRADIENT ESTIMATES OF A NONLINEAR ELLIPTIC EQUATION FOR THE V -LAPLACIAN

  • Zeng, Fanqi
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.853-865
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    • 2019
  • In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $${\Delta}_Vu+cu^{\alpha}=0$$, where c, ${\alpha}$ are two real constants and $c{\neq}0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-${\acute{E}}mery$ Ricci curvature bounded from below, which generalize some results of [8].

MULTIPLICITY RESULTS FOR SOME FOURTH ORDER ELLIPTIC EQUATIONS

  • Jin, Yinghua;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.18 no.4
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    • pp.489-496
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    • 2010
  • In this paper we consider the Dirichlet problem for an fourth order elliptic equation on a open set in $R^N$. By using variational methods we obtain the multiplicity of nontrivial weak solutions for the fourth order elliptic equation.

Preconditioning Cubic Spline Collocation Methods for a Coupled Elliptic Equation

  • Shin, Byeong-Chun;Kim, Sang-Dong
    • Kyungpook Mathematical Journal
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    • v.50 no.3
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    • pp.419-431
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    • 2010
  • A low-order finite element preconditioner is analyzed for a cubic spline collocation method which is used for discretizations of coupled elliptic problems derived from an optimal control problrm subject to an elliptic equation. Some numerical evidences are also provided.

A CERTAIN EXAMPLE FOR A DE GIORGI CONJECTURE

  • Cho, Sungwon
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.763-769
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    • 2014
  • In this paper, we illustrate a counter example for the converse of a certain conjecture proposed by De Giorgi. De Giorgi suggested a series of conjectures, in which a certain integral condition for singularity or degeneracy of an elliptic operator is satisfied, the solutions are continuous. We construct some singular elliptic operators and solutions such that the integral condition does not hold, but the solutions are continuous.

EXISTENCE OF INFINITELY MANY SOLUTIONS OF THE NONLINEAR HIGHER ORDER ELLIPTIC EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.309-322
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    • 2008
  • We prove the existence of infinitely many solutions of the nonlinear higher order elliptic equation with Dirichlet boundary condition $(-{\Delta})^mu=q(x,u)$ in ${\Omega}$, where $m{\geq}1$ is an integer and ${\Omega}{\subset}{R^n}$ is a bounded domain with smooth boundary, when q(x,u) satisfies some conditions.

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