• Korean Journal of Mathematics
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• v.18 no.2
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• pp.201-211
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• 2010

We investigate the number of the nontrivial solutions of the nonlinear biharmonic equation with Dirichlet boundary condition. We give a theorem that there exist at least three nontrivial solutions for the nonlinear biharmonic problem. We prove this result by the finite dimensional reduction method and the shape of the graph of the corresponding functional on the finite reduction subspace.

• Journal for History of Mathematics
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• v.16 no.4
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• pp.107-115
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• 2003

The nonlinear fourth order elliptic equations with jumping nonlinearity was modeled by McKenna. We investigate the trends for the researches of the existence of solutions of a fourth order semilinear elliptic boundary value problem with Dirichlet boundary Condition, ${\Delta}^2u{＋}c{\Delta}u=b_1[(u＋1)^{-}1]{＋}b_2u^+$ in ${\Omega}$, where ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.707-720
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• 2014

We get a theorem which shows the existence of at least three solutions for some elliptic system with Dirichlet boundary condition. We obtain this result by using the finite dimensional reduction method which reduces the infinite dimensional problem to the finite dimensional one. We also use the critical point theory on the reduced finite dimensioal subspace.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.155-167
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• 2019

In this article we are concerned with some non-local problems of Kirchhoff type with Dirichlet boundary condition in Orlicz-Sobolev spaces. A result of the existence of infinitely many solutions is established using variational methods and Ricceri's critical points principle modified by Bonanno.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.871-884
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• 2000

Polyharmonic operator with Dirichlet boundary condition is considered in a n-dimensional cone. The regularity properties of weak solutions are studied. In particular, it is proved the Holder contionuity of solutions near the vertex of the cone for dimensions n=2m+3,2m+4, where 2m is the order of the polyharmonic operator.

• 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.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.1059-1059
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• 2020

• Journal of applied mathematics & informatics
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• v.39 no.1_2
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• pp.165-172
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• 2021

We prove a lower estimate of Neumann eigenvalues on compact manifolds with the condition that the Ricci curvature is bounded below. We improved the earlier results.

• Proceedings of the KSME Conference
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• 2001.06e
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• pp.506-511
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• 2001

This paper presents how redevelopment of the boundary layer in a backward-facing step flow is affected by boundary conditions imposed on velocity at the inlet, top and exit of the flow. A two-dimensional, laminar, incompressible flow over a backward-facing step with an open top boundary has been computed by using numerical methods of second-order time and spatial accuracy and a fractional-step method that guarantees a divergence-free velocity field at all time. The inlet velocity profile above the step is of Blasius type. Along the top boundary, shear-tree and Dirichlet conditions on the streamwise velocity were considered and at the exit fully-developed and convective boundary conditions were examined. (The vertical velocity at all boundaries were assumed to be zero explicitly or implicitly.) From the computed flow fields, the reattachment on the bottom side of shear layer separated from the tip of the step and succeeding redevelopment of the boundary layer were investigated.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.339-345
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• 2008

We prove the existence of the positive solution for the nonlinear system of suspension bridge equations with Dirichlet boundary condition and periodic condition $$\{u_{tt}+u_{xxxx}+av^+=1+{\epsilon}_1h_1(x,t)\text{ in }(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,\\v_{tt}+v_{xxxx}+bu^+=1+{\epsilon}_2h_2(x,t)\text{ in }(-\frac{\pi}{2},\frac{\pi}{2}){\times}R,$$ where $u^+={\max}\{u,0\},\;{\epsilon}_1,\;{\epsilon}_2$ are small numbers and $h_1(x,t)$, $h_2(x,t)$ are bounded, ${\pi}$-periodic in t and even in x and t and ${\parallel}h_1{\parallel}={\parallel}h_2{\parallel}=1$.