• 제목/요약/키워드: mountain pass method

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EXISTENCE OF THE SOLUTIONS FOR THE ELLIPTIC PROBLEM WITH NONLINEAR TERM DECAYING AT THE ORIGIN

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • 제20권4호
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    • pp.533-540
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    • 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.

$L_q$ estimation on the least energy solutions

  • Pahk, Dae-Hyeon;Park, Sang-Don
    • 대한수학회지
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    • 제32권2호
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    • pp.329-339
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    • 1995
  • Let us consider the Neumann problem for a quasilinear equation $$ (I_\varepsilon) {\varepsilon^m div($\mid$\nabla_u$\mid$^{m-2}\nabla_u) - u$\mid$u$\mid$^{m-2} + f(u) = 0 in \Omega {\frac{\partial\nu}{\partial u} = 0 on \partial\Omega. $$ where $1 < m < N, N \geq 2, \varepsilon > 0, \Omega$ is a smooth bounded domain in $R^n$ and $\nu$ is the unit outer normal vector to $\partial\Omega$.

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ON SOME p(x)-KIRCHHOFF TYPE EQUATIONS WITH WEIGHTS

  • Chung, Nguyen Thanh
    • Journal of applied mathematics & informatics
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    • 제32권1_2호
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    • pp.113-128
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    • 2014
  • Consider a class of p(x)-Kirchhoff type equations of the form $$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$ where p(x), $q(x){\in}C({\bar{\Omega}})$ with 1 < $p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$ < N, $M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$ is a continuous function that may be degenerate at zero, ${\lambda}$ is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function V (x) may change sign or not.

ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

  • Bin, Ge
    • 대한수학회보
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    • 제51권2호
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    • pp.409-421
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    • 2014
  • This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $$\{-div\((1+\frac{|{\nabla}u|^{p(x)}}{\sqrt{1+|{\nabla}u|^{2p(x)}}}})|{\nabla}u|^{p(x)-2}{\nabla}u\)={\lambda}f(x,u)\;a.e.\;in\;{\Omega}\\u=0,\;on\;{\partial}{\Omega}$$ where ${\Omega}{\subset}R^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\lambda}$ > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter ${\lambda}$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter ${\lambda}$ > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.

ON EXISTENCE OF WEAK SOLUTIONS OF NEUMANN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING p-LAPLACIAN IN AN UNBOUNDED DOMAIN

  • Hang, Trinh Thi Minh;Toan, Hoang Quoc
    • 대한수학회보
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    • 제48권6호
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    • pp.1169-1182
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    • 2011
  • In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+b(x){\mid}u{\mid}^{p-2}u=f(x,\;u),\;p{\geq}2$$ in an unbounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, with sufficiently smooth bounded boundary ${\partial}{\Omega}$, where $h(x){\in}L_{loc}^1(\overline{\Omega})$, $\overline{\Omega}={\Omega}{\cup}{\partial}{\Omega}$, $h(x){\geq}1$ for all $x{\in}{\Omega}$. The proof of main results rely essentially on the arguments of variational method.

기존선 철도구조물 접속부의 보강에 대한 고찰 (A Study of Reinforcement of Railway Structure Approaches in Conventional line)

  • 박준오;이상배;김관형
    • 한국철도학회:학술대회논문집
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    • 한국철도학회 2007년도 추계학술대회 논문집
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    • pp.442-452
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    • 2007
  • Korean trains pass many mountain areas, so the volume of structures like bridge and tunnel has large part of railway lines. Train speed-up naturally needs a straight line in railway, then structures are increasing, and this influences passenger's comfort and the safety of operation, and it needs more track maintenance. The stiffness of bridge and tunnel is higher than the soil in the roadbed in spite of dynamic difference in vibration and displacement. Differences in stiffness have more dynamic effects and increase the deformation and destruction in the track and roadbed. This study will measure periodically to structure's approaches which have very fast track irregularity and analyze dynamic differences and track irregularity near structure's approaches, so realize the cause of track irregularity of structure's approaches and use basic data for reasonably strengthening method of structure's approaches.

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SOLUTIONS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEM WITH MIXED NONLINEARITIES

  • Zhang, Ziheng
    • 대한수학회보
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    • 제53권5호
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    • pp.1585-1596
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    • 2016
  • In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem (FBVP) $$\{_tD_T^{\alpha}(_0D_t^{\alpha}u(t))={\nabla}W(t,u(t)),\;t{\in}[0,T],\\u(0)=u(T)=0,$$ where ${\alpha}{\in}(1/2,1)$, $u{\in}{\mathbb{R}}^n$, $W{\in}C^1([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ and ${\nabla}W(t,u)$ is the gradient of W(t, u) at u. The novelty of this paper is that, when the nonlinearity W(t, u) involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.

현장계측을 통한 기존선 철도 구조물 접속부의 거동분석 (Dynamic Analysis of Structure's Approaches through Field Tests in the Conventional Railway)

  • 박준오;이상배;홍원표
    • 한국철도학회:학술대회논문집
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    • 한국철도학회 2007년도 춘계학술대회 논문집
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    • pp.1864-1874
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    • 2007
  • Korean trains pass many mountain areas, so the volume of structures like bridge and tunnel has large part of railway lines. Train speed-up naturally needs a straight line in railway, then structures are increasing, and the length of structure has more than 70% in Kyongbu high-speed railway. The stiffness of bridge and tunnel is higher than the soil in the roadbed in spite of dynamic difference in vibration and displacement. Differences in stiffness have more dynamic effects and increase the deformation and destruction in the track and roadbed. This influences passenger's comfort and the safety of operation, and it needs more track maintenance. This study selected tunnel with ballast track, tunnel with concrete track, and structure's approaches with short maintenance cycle in the roadbed and had track acceleration tests and track liner inspections using track master in the field. This study will measure periodically to structure's approaches which have very fast track irregularity and analyze dynamic differences and track irregularity near structure's approaches, so realize the cause of track irregularity of structure's approaches and use basic data for reasonably strengthening method of structure's approaches.

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Glass Dissolution Rates From MCC-1 and Flow-Through Tests

  • Jeong, Seung-Young
    • 한국방사성폐기물학회:학술대회논문집
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    • 한국방사성폐기물학회 2004년도 학술논문집
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    • pp.257-258
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    • 2004
  • The dose from radionuclides released from high-level radioactive waste (HLW) glasses as they corrode must be taken into account when assessing the performance of a disposal system. In the performance assessment (PA) calculations conducted for the proposed Yucca Mountain, Nevada, disposal system, the release of radionuclides is conservatively assumed to occur at the same rate the glass matrix dissolves. A simple model was developed to calculate the glass dissolution rate of HLW glasses in these PA calculations [1]. For the PA calculations that were conducted for Site Recommendation, it was necessary to identify ranges of parameter values that bounded the dissolution rates of the wide range of HLW glass compositions that will be disposed. The values and ranges of the model parameters for the pH and temperature dependencies were extracted from the results of SPFT, static leach tests, and Soxhlet tests available in the literature. Static leach tests were conducted with a range of glass compositions to measure values for the glass composition parameter. The glass dissolution rate depends on temperature, pH, and the compositions of the glass and solution, The dissolution rate is calculated using Eq. 1: $rate{\;}={\;}k_{o}10^{(ph){\eta})}{\cdot}e^{(-Ea/RT)}{\cdot}(1-Q/K){\;}+{\;}k_{long}$ where $k_{0},\;{\eta}$ and Eaare the parameters for glass composition, pH, $\eta$ and temperature dependence, respectively, and R is the gas constant. The term (1-Q/K) is the affinity term, where Q is the ion activity product of the solution and K is the pseudo-equilibrium constant for the glass. Values of the parameters $k_{0},\;{\eta}\;and\;E_{a}$ are the parameters for glass composition, pH, and temperature dependence, respectively, and R is the gas constant. The term (1-Q/C) is the affinity term, where Q is the ion activity product of the solution and K is the pseudo-equilibrium constant for the glass. Values of the parameters $k_0$, and Ea are determined under test conditions where the value of Q is maintained near zero, so that the value of the affinity term remains near 1. The dissolution rate under conditions in which the value of the affinity term is near 1 is referred to as the forward rate. This is the highest dissolution rate that can occur at a particular pH and temperature. The value of the parameter K is determined from experiments in which the value of the ion activity product approaches the value of K. This results in a decrease in the value of the affinity term and the dissolution rate. The highly dilute solutions required to measure the forward rate and extract values for $k_0$, $\eta$, and Ea can be maintained by conducting dynamic tests in which the test solution is removed from the reaction cell and replaced with fresh solution. In the single-pass flow-through (PFT) test method, this is done by continuously pumping the test solution through the reaction cell. Alternatively, static tests can be conducted with sufficient solution volume that the solution concentrations of dissolved glass components do not increase significantly during the test. Both the SPFT and static tests can ve conducted for a wide range of pH values and temperatures. Both static and SPFt tests have short-comings. the SPFT test requires analysis of several solutions (typically 6-10) at each of several flow rates to determine the glass dissolution rate at each pH and temperature. As will be shown, the rate measured in an SPFt test depends on the solution flow rate. The solutions in static tests will eventually become concentrated enough to affect the dissolution rate. In both the SPFt and static test methods. a compromise is required between the need to minimize the effects of dissolved components on the dissolution rate and the need to attain solution concentrations that are high enough to analyze. In the paper, we compare the results of static leach tests and SPFT tests conducted with simple 5-component glass to confirm the equivalence of SPFT tests and static tests conducted with pH buffer solutions. Tests were conducted over the range pH values that are most relevant for waste glass disssolution in a disposal system. The glass and temperature used in the tests were selected to allow direct comparison with SPFT tests conducted previously. The ability to measure parameter values with more than one test method and an understanding of how the rate measured in each test is affected by various test parameters provides added confidence to the measured values. The dissolution rate of a simple 5-component glass was measured at pH values of 6.2, 8.3, and 9.6 and $70^{\circ}C$ using static tests and single-pass flow-through (SPFT) tests. Similar rates were measured with the two methods. However, the measured rates are about 10X higher than the rates measured previously for a glass having the same composition using an SPFT test method. Differences are attributed to effects of the solution flow rate on the glass dissolution reate and how the specific surface area of crushed glass is estimated. This comparison indicates the need to standardize the SPFT test procedure.

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