• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1041-1054
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• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.291-307
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• 1996

In this paper, we shall show the existence of solutions of the following nonlinear partial differential equation $$ {^{divA(-\Delta u) = f(x, u, \Delta u) in \Omega}^{u = 0 on \partial\Omega} $$ where $f(x, u, \Delta u) = -u$\mid$\Delta u$\mid$^{p-2} + h, p \geq 2, h \in L^\infty$.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.771-783
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• 2018

This paper, we investigate a strongly damped nonlinear Klein-Gordon equation with nonlinearities of variable exponent type $$u_{tt}-{\Delta}u-{\Delta}u_t+m^2u+{\mid}u_t{\mid}^{p(x)-2}u_t={\mid}u{\mid}^{q(x)-2}u$$ associated with initial and Dirichlet boundary conditions in a bounded domain. We obtain a nonexistence of solutions if variable exponents p (.), q (.) and initial data satisfy some conditions.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

• Resources Recycling
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• v.22 no.3
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• pp.57-64
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• 2013

Sulfurization reaction characteristics of $Eu_2O_3$, uranium oxides($UO_2$, $U_3O_8$), mixture of $Eu_2O_3$ and uranium oxides, Eu-doped uranium oxides($(U,Eu)O_2$, $(U,Eu)_3O_8$), and phase-separated products prepared by HOX (High temperature OXidation) of $(U,Eu)O_2$ were investigated in the temperature range from 400 to $800^{\circ}C$. Only $Eu_2O_3$ in the mixture of $Eu_2O_3$ and uranium oxides was converted into $Eu_3S_4$ by sulfurization reaction at $450^{\circ}C$ without reaction between them. Sulfurization reaction behavior of $(U,Eu)_3O_8$ and $(U,Eu)O_2$ up to $600^{\circ}C$ was similar to $U_3O_8$ and $UO_2$, respectively, while they were sulfurized into Eu-rich $(U,Eu)S_x$ and ${\alpha}-US_2$ at $800^{\circ}C$. In the sulfurization of RE-rich $(U,Eu)_4O_9$ and $U_3O_8$ prepared by high temperature oxidation, it was confirmed that RE-rich $(U,Eu)S_x$ and UOS phases were formed at $600^{\circ}C$. For Eu-rich $(U,Eu)O_2$ and $UO_2$ prepared by reduction of HOX products, it was identified that Eu-rich (U,Eu)OS was formed at $450^{\circ}C$ by sulfurization of Eu-rich $(U,Eu)O_2$, while $UO_2$ remained unreacted.

• Communications of the Korean Mathematical Society
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• v.16 no.2
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• pp.287-290
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• 2001

Let X$_1$, X$_2$, … be a sequence of independent and identically distributed random variables with continuous cumulative distribution function F(x). X(sub)j is an upper record value of this sequence if X(sub)j > max {X$_1$, X$_2$, …, X(sub)j-1}. We define u(n) = min {j│j > u(n-1), X(sub)j > X(sub)u(n-1), n $\geq$ 2} with u(1) = 1. Then F(x) = 1 - e(sup)-x/c, x > 0 if and only if E[X(sub)n(n+1) - X(sub)u(n)│X(sub)u(m) = y] = c or E[X(sub)u(n+2) - X(sub)u(n)│X(sub)u(m) = y] = 2c, n $\geq$ m+1.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1551-1571
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• 2020

This paper is concerned with the following Kirchhoff-Schrödinger-Poisson system $$\begin{cases} -(a+b{\displaystyle\smashmargin{2}\int\nolimits_{\mathbb{R}^3}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+{\mu}{\phi}u={\lambda}f(x){\mid}u{\mid}^{p-2}u+g(x){\mid}u{\mid}^{p-2}u,&{\text{ in }}{\mathbb{R}}^3,\\-{\Delta}{\phi}={\mu}{\mid}u{\mid}^2,&{\text{ in }}{\mathbb{R}}^3, \end{cases}$$ where a > 0, b, µ ≥ 0, p ∈ (1, 2), q ∈ [4, 6) and λ > 0 is a parameter. Under some suitable assumptions on V (x), f(x) and g(x), we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.67-86
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• 2014

Based on an expanded mixed finite element method, we consider the semidiscrete approximations of the solution u of the quasilinear pseudo-parabolic equation defined on ${\Omega}{\subset}R^d$, $1{\leq}d{\leq}3$. We construct the semidiscrete approximations of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u and prove the existence of the semidiscrete approximations. And also we prove the optimal convergence of ${\nabla}u$ and $a(u){\nabla}u+b(u){\nabla}u_t$ as well as u in $L^2$ normed space.

• Korean Journal of Mathematics
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• v.17 no.2
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• pp.197-210
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• 2009

We prove the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the elliptic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}A{\xi}+g_1({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\\A{\xi}+g_2({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\end{array}$$ where $lim_{u{\rightarrow}{\infty}}\frac{gj(u)}{u}={\beta}_j$, $lim_{u{\rightarrow}-{\infty}}\frac{gj(u)}{u}={\alpha}_j$ are finite and the nonlinearity $g_1+2g_2$ crosses eigenvalues of A.

• Korean Journal of Mathematics
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• v.21 no.2
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• pp.203-212
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• 2013

We investigate the existence of solutions of the one-dimensional wave system $$u_{tt}-u_{xx}+{\mu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\v_{tt}-v_{xx}+{\nu}g(u+v)=f(u+v)\;\;in\;(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ with Dirichlet boundary condition. We find them by applying linking inequlaities.