• Korean Journal of Mathematics
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• 제19권4호
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• pp.423-436
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• 2011

We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

• 대한수학회지
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• 제56권4호
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• pp.857-867
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• 2019

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

• East Asian mathematical journal
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• 제40권3호
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• pp.295-306
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• 2024

We investigate the existence of multiple positive solutions of the following elliptic equation with a Schrödinger-type term: $$\begin{cases}-{\Delta}u+V(x)u={\lambda}f(u){\quad} x{\in}{\Omega},\\{\qquad}{\qquad}{\quad}u=0, {\qquad}\;x{\in}\partial{\Omega},\end{cases}$$, where 0 ∈ Ω is a bounded domain in ℝ^N , N ≥ 1, with a smooth boundary ∂Ω, f ∈ C[0, ∞), V ∈ L^∞(Ω) and λ is a positive parameter. In particular, when f(s) > 0 for 0 ≤ s < σ and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

• 대한수학회지
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• 제37권5호
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• pp.751-761
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• 2000

Symmetry properties of positive solutions for semilinear elliptic problems in n are considered. We give a symmetry result for the problem in the feneral case, and then derive various results for certain classes of demilinear elliptic equations. We employ the moving plane method based on the maximum principle on unbounded domains to obtain the result on symmetry.

• Journal of Astronomy and Space Sciences
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• 제24권4호
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• pp.315-326
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• 2007

최근에 제안된, 이웃하는 타원 궤도의 상대운동에 대한 몇 가지 양함수형 해를 분석하였다. 이 해를 이용한 상대운동 결과를 일반 선형화 운동 방정식의 해석적 해와 비교했다. 수치계산 결과를 위한 초기 조건은 Hill-Clohessy-Wiltshire(HCW) 운동방정식에 의한 해의 역함수로 구했다. 기준 궤도의 차이에도 불구하고 상대적으로 작은 이심률의 궤도 일 경우에는 타원 상대운동 궤도와 원 상대운동 궤도의 결과는 근접했다. 주위성의 궤도가 상대적으로 큰 이심률을 가질 경우에는, 기본 궤도로 원 궤도를 이용하는 HCW 운동방정식은 다른 타원 상대운동 궤도 방정식의 해보다 상대적으로 큰 오차를 갖는다.

• 한국수학사학회지
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• 제16권1호
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• pp.17-24
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• 2003

The Pythagorean equation $x^2{＋}y^2{=}z^2$ and Pythagorean triple had appeared in the Babylonian clay tablet made between 1900 and 1600 B. C. Another quadratic equation called Pell equation was implicit in an Archimedes' letter to Eratosthenes, so called ‘cattle problem’. Though elliptic equation were contained in Diophantos’ Arithmetica, a substantial progress for the solution of cubic equations was made by Bachet only in 1621 when he found infinitely many rational solutions of the equation $y^2{=}x^3{-}2$. The equation $y^2{=}x^3{+}c$ is the simplest of all elliptic equations, even of all Diophantine equations degree greater than 2. It is due to Bachet, Dirichlet, Lebesque and Mordell that the equation in better understood.

• 한국수학사학회지
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• 제15권3호
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• pp.83-93
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• 2002

We investigate a history of reseahches of a nonlinear elliptic equation with jumping nonlinearity, under Dirichlet boundary condition. The investigation will be focussed on the researches by topological methods. We also add recent researches, relations between multiplicity of solutions and source terms of tile equation when the nonlinearity crosses two eigenvalues and the source term is generated by three eigenfunctions.

• 대한수학회지
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• 제58권5호
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• pp.1279-1298
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• 2021

The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate hyperbolic equation involving strongly degenerate elliptic differential operators. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

• 한국해안해양공학회지
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• 제6권1호
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• pp.81-87
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• 1994

완경사 방정식을 에너지 보존식으로부터 직접 유도하였으며 에너지 보존식과 Green's first and second identities와의 관계를 분명히 밝혔다. 파랑-흐름 상호작용 시의 타원형 완경사 방정식이 Berkhoff (1972)에 의해서 유도된 흐름이 없는 경우와 같은 형태를 갖는다는 것이 제시되었으며, 또한 해석적인 해를 통해 물리적 특성이 조사되었다.

• 대한수학회지
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• 제57권6호
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• pp.1509-1533
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• 2020

We provide detailed proofs for local gradient estimates for elliptic equations in divergence form with partial Dini mean oscillation coefficients in a ball and a half ball.