• Korean Journal of Mathematics
• /
• v.26 no.1
• /
• pp.9-21
• /
• 2018

We investigate existence and multiplicity of the solutions for elliptic boundary value problem with two singularities. We obtain one theorem which shows that there exists at least one nontrivial weak solution under some conditions on which the corresponding functional of the problem satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.

• East Asian mathematical journal
• /
• v.35 no.1
• /
• pp.41-50
• /
• 2019

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

• Journal of the Korean Mathematical Society
• /
• v.57 no.6
• /
• pp.1451-1470
• /
• 2020

We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)^s_p is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L^∞(Ω) of any possible weak solution by applying the bootstrap argument.

• Korean Journal of Mathematics
• /
• v.19 no.3
• /
• pp.331-342
• /
• 2011

We consider the multiplicity of the solutions for a class of a system of critical growth beam equations with periodic condition on t and Dirichlet boundary condition $$\{u_{tt}+u_{xxxx}=av+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{{\alpha}-1}v_{+}^{\beta}+s{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,\\u_{tt}+v_{xxxx}=bu+\frac{2{\alpha}}{{\alpha}+{\beta}}u_{+}^{\alpha}v_{+}^{{\beta}-1}+t{\phi}_{00}\;\;in\;(-\frac{\pi}{2},\;\frac{\pi}{2}){\times}R,$$ where ${\alpha}$, ${\beta}$ > 1 are real constants, $u_+=max\{u,0\}$, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_00=1$ of the eigenvalue problem $u_{tt}+u_{xxxx}={\lambda}_{mn}u$. We show that the system has a positive solution under suitable conditions on the matrix $A=\(\array{0&a\\b&0}\)$, s > 0, t > 0, and next show that the system has another solution for the same conditions on A by the linking arguments.

• Journal of Aerospace System Engineering
• /
• v.4 no.1
• /
• pp.1-9
• /
• 2010

As the rocket KSLV-1 called NARO was launched lately, development of domestic rocket technology has been accelerated elastically. Since the rocket technology needs a lot of empirical data, a variety of experiments should be done and lots of time have to be spent for accumulating the foundation of technology. However using a computer can be the solution to close a gap of technique because the simulation can be executed in short time against real experiments and calculate a multiplicity of cases easily. In this research, the transient analysis of turbopump-fed liquid rocket system was worked by the one dimensional modeling. The rocket system consists of the modulized components that are engine, turbopump and so on. For 70 ton class system, the rocket transient process of starting was studied and the performance analysis in steady condition was achieved. In addition, the estimation of nozzle internal flow was investigated by using a nozzle coefficient.

• Journal of the Korean Physical Society
• /
• v.73 no.10
• /
• pp.1512-1518
• /
• 2018

The problem of scalar wave scattering by a sphere on or near a planar substrate is analytically solved. The solution is a set of wave functions coming in the form of infinite series of spherical and plane waves. In air, the incident plane wave is either scattered by the sphere or reflected from the substrate. A part of these scattered or reflected waves propagate to the other object where it is reflected and scattered again. Such processes of scattering and reflection repeat in turn indefinitely to generate multiply scattered waves, which are represented in the corresponding terms in the infinite series. The term in the series can be arranged in a recognizable manner to explicitly reveal the involved process and the multiplicity of scattering.

• Journal of applied mathematics & informatics
• /
• v.28 no.3_4
• /
• pp.697-710
• /
• 2010

In this paper, we prove existence of infinitely many positive and concave solutions, by means of a simple approach, to $3^{th}$ order three-point singular boundary value problem {$x^{\prime\prime\prime}(t)=\alpha(t)f(t,x(t))$, 0 < t < 1, $x(0)=x'(\eta)=x^{\prime\prime}(1)=0$, (1/2 < $\eta$ < 1). Moreover with respect to multiplicity of solutions, we don't assume any monotonicity on the nonlinearity. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Knesser's type properties of the solutions funnel and the well-known Krasnosel'ski$\breve{i}$'s fixed point theorem. The later is applied on a new very simple cone K, just on the plane $R^2$. These extensions justify the efficiency of our new approach compared to the commonly used one, where the cone $K\;{\subset}\;C$ ([0, 1], $\mathbb{R}$) and the existence of a positive Green's function is a necessity.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.33 no.5
• /
• pp.447-454
• /
• 2009

Rate-independent crystal plasticity model suffers from the non-uniqueness of activated slip systems and the determination of the shear slip rates on the active slip systems. In this paper, a time-integration algorithm which circumvents the problem of the multiplicity of the slip systems was developed and implemented into the user subroutine VUMAT of a commercial finite element program ABAQUS. The magnitude of the slip shears on the active slip systems in f.c.c Cu single crystal aligned with the specific crystallographic orientation was investigated to validate our solution procedure. Also, texture developments under various deformation modes such as simple compression, simple tension and plane strain compression were compared with the results of the rate-dependent model by using the rate-independent crystal plasticity model. The computation time employing the rate-independent model is much more reduced than the those of the rate-dependent model.

• 대한공업교육학회지
• /
• v.30 no.2
• /
• pp.123-129
• /
• 2005

A Rayleigh-Benard problem with non-uniform wall temperatures of the form, $T_L=T_1+{\delta}{\Delta}T{\sin}kx$ and $T_U=T_2-{\delta}{\Delta}{\sin (kx)$, is numerically investigated. In the conduction-dominated regime with small a Rayleigh number, a two-tier structure appears with two counter-rotating rolls stacked on the top of each other. The flow becomes unstable with increase of the Rayleigh number, and multicellular convection occurs above a critical Rayleigh number. The multicellular flows at high Rayleigh numbers consist of approximetely square-shape cells. Four multiple flows and dual flows classified by the number of cells are found at k=0.5 and k=1, respectively.

• Korean Journal of Mathematics
• /
• v.20 no.4
• /
• pp.533-540
• /
• 2012

We consider the multiplicity of the solutions for the elliptic boundary value problem with $C^1$ nonlinear term decaying at the origin. We get a theorem which shows the existence of the nontrivial solution for the elliptic problem with $C^1$ nonlinear term decaying at the origin. We obtain this result by reducing the elliptic problem with the $C^1$ nonlinear term to the el-liptic problem with bounded nonlinear term and then approaching the variational method and using the mountain pass geometry for the reduced the elliptic problem with bounded nonlinear term.