• Journal of the Korean Mathematical Society
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• v.37 no.1
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• pp.85-109
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• 2000

In this paper, the multiplicity, stability and the structure of classical solutions of semilinear elliptic equations of the form (equation omitted) will be discussed. Here $\Omega$ is a smooth and bounded domain in $R^{n}$ (n $\geq$ 1), f(x,u) = │u│$^{\alpha}$/sgn(u)-h(x), 0 < $\alpha$ < 1, (n $\geq$ 1) and h is a ${\gamma}$- Holder continuous function on $\Omega$ for some 0 < ${\gamma}$ < 1.a}$ < 1.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.1011-1016
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• 2011

In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equation on $R^N$. we obtained the results under different suitable conditions on the locally H$\"{o}$lder continuous nonlinearity f(x, u), we needn't any mono-tonicity condition about the nonlinearity.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1577-1586
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• 2018

In this paper, we prove that if every bounded ${\mathcal{A}}$-harmonic function on a complete Riemannian manifold M is asymptotically constant at infinity of p-nonparabolic ends of M, then each bounded ${\mathcal{A}}$-harmonic function is uniquely determined by the values at infinity of p-nonparabolic ends of M, where ${\mathcal{A}}$ is a nonlinear elliptic operator of type p on M. Furthermore, in this case, every bounded ${\mathcal{A}}$-harmonic function on M has finite energy.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.259-271
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• 2006

We consider a semilinear elliptic boundary value problem with Dirichlet boundary condition $Au+bu^+-au^-=t_{1{\phi}1}+t_{2{\phi}2}$ in ${\Omega}$ and ${\phi}_n$ is the eigenfuction corresponding to ${\lambda}_n(n=1,2,{\cdots})$. We have a concern with the multiplicity of solutions of the equation when ${\lambda}_1$ < a < ${\lambda}_2$ < b < ${\lambda}_3$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.2
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• pp.215-227
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• 2020

A new multiscale finite element method for elliptic problems with highly oscillating coefficients are introduced. A hybridization yields a locally flux-conserving numerical scheme for multiscale problems. Our approach naturally induces a homogenized equation which facilitates error analysis. Complete convergence analysis is given and numerical examples are presented to validate our analysis.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.73-79
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• 2008

The purpose of this paper is to study the relations between quasilinear elliptic equations on Riemannian manifolds and differential forms. Two classes of differential forms are introduced and it is shown that some differential expressions are connected in a natural way to quasilinear elliptic equations.

• Korean Journal of Mathematics
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• v.26 no.4
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• pp.747-756
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• 2018

We study a maximum principle for a uniformly elliptic second order differential operator in nondivergence form. We allow a bounded positive zeroth order coefficient in a certain type of unbounded domains. The results extend a result by J. Busca in a sense of domains, and we present a simple proof based on local maximum principle by Gilbarg and Trudinger with iterations.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.24 no.4
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• pp.355-364
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• 2011

In this study, the details of WSA(wavelet series analysis) have been demonstrated to solve the 4th-order elliptic differential equation. It is clear to solve the 2nd-order elliptic differential equation with the basis function of Hat wavelet series that is used in the previous study existed in $H^1$-space. However, it is difficult to solve the 4th order differential equation with same basis function of Hat wavelet series because of insufficient differentiability and integrability. To overcome this problem, the linear equations in terms of moment and deflection have been formulated and solved sequentially that are similar to extension of Elastic Load Method and Moment Area Method in some senses. Also, the differences and common points between the proposed method and the meshless method are discussed in the procedure of WSA formulation. As we expect, it is easy to ascertain that the more terms of Hat wavelet series are used, the better numerical solutions are improved. Also the solutions obtained by WSA have been compared with the conventional FEM solutions in case of Euler beam problems with stress singularity.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.261-269
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• 2008

Let $B_1$ be a unit ball in $R^n(n{\geq}3)$, and $2^*=2n/(n-2)$ be the critical Sobolev exponent for the embedding $H_0^1(B_1){\hookrightarrow}L^{2^*}(B_1)$. By using a variant of Pohoz$\check{a}$aev's identity, we prove the nonexistence of nodal solutions for the Dirichlet problem $-{\Delta}u-{\mu}\frac{u}{{\mid}x{\mid}^2}={\lambda}u+{\mid}u{\mid}^{2^*-2}u$ in $B_1$, u=0 on ${\partial}B_1$ for suitable positive numbers ${\mu}$ and ${\nu}$.

• Journal of Ocean Engineering and Technology
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• v.22 no.5
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• pp.39-43
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• 2008

For a partially filled and deeply submersed membrane container, an analytic solution for similarity shape was studied. The static shape of a membrane container can be expressed as a set of nonlinear ordinary differential equations. These equations are combined into an integrable equation. The solution of the equation is derived in terms of elliptic integrals, the arguments of which contain an unknown at the point of inflection. The point of inflection is determined by using the boundary condition at a separating point. Some characteristic values of the similarity shape were evaluated and the shapes are illustrated.