• Honam Mathematical Journal
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• v.27 no.4
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• pp.609-619
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• 2005

We are concerned with the multiplicity of solutions of the nonlinear elliptic equation with Dirichlet boundary condition. We reveal the existence of m solutions of the nonlinear elliptic equation by a critical point theory, under some condition.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.301-310
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• 1999

This paper is concerned with the existence of positive solution of a semilinear elliptic equation with homogeneous Dirichlet boundary condition.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.4
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• pp.231-244
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• 2023

We study the modularity and holomorphic anomaly equation for Hodge-Gromov-Witten invariants of elliptic curves.

• 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].

• 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.

• 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.

• 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.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.111-127
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• 2011

We prove the uniqueness solvability of the Tricomi problem for the elliptic - hyperbolical equation of the second type by using a new representation of the solution in the generalized class R.

• 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.