• Title/Summary/Keyword: ground state solution

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POSITIVE SOLUTION AND GROUND STATE SOLUTION FOR A KIRCHHOFF TYPE EQUATION WITH CRITICAL GROWTH

  • Chen, Caixia;Qian, Aixia
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.961-977
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    • 2022
  • In this paper, we consider the following Kirchhoff type equation on the whole space $$\{-(a+b{\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}\;{\mid}{\nabla}u{\mid}^2dx){\Delta}u=u^5+{\lambda}k(x)g(u),\;x{\in}{\mathbb{R}}^3,\\u{\in}{\mathcal{D}}^{1,2}({\mathbb{R}}^3),$$ where λ > 0 is a real number and k, g satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

Intramolecular Proton Transfers of 2-hydroxy-4,5-naphthotropone

  • Du-Jeon Jang
    • Bulletin of the Korean Chemical Society
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    • v.12 no.4
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    • pp.441-444
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    • 1991
  • The intramolecular proton transfers of 2-hydroxy-4,5-naphthotropone in room temperature solutions are studied using static and time-resolved absorption and emission spectroscopy. Dual normal and tautomer fluorescence is observed in ethanol solution, while only the tautomer fluorescence is observed in cyclohexane solution. The fluorescence lifetimes and quantum yields in ethanol and cyclohexane solutions indicate that in hydrocarbon solvents, rapid intersystem crossing competes with proton transfer in the first excited singlet state. Transient absorption spectra and kinetics indicate that proton transfer also undergoes in the first triplet state with a transfer time of ∼ 3 ns. No transient absorption from the tautomer ground state indicates a rapid back proton transfer in the ground state.

GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER-POISSON-KIRCHHOFF TYPEPROBLEMS WITH A CRITICAL NONLINEARITY IN ℝ3

  • Qian, Aixia;Zhang, Mingming
    • Journal of the Korean Mathematical Society
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    • v.58 no.5
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    • pp.1181-1209
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    • 2021
  • In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff system $$\;\{\begin{array}{lll}-(1+b{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+k(x){\phi}u={\lambda}f(x)u+{\mid}u{\mid}^4u,&&\text{in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=k(x)u^2,&&\text{in }{\mathbb{R}}^3,\end{array}$$ where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem -∆u + V(x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.

Evaluation of Tunnel Face Stability with the Consideration of Seepage Forces (침투력을 고려한 토사터널 막장의 안정성 평가방법에 대한 고찰)

  • 남석우;이인모
    • Proceedings of the Korean Geotechical Society Conference
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    • 1999.10a
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    • pp.193-200
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    • 1999
  • Since Broms and Bennermark(1967) suggested the face stability criterion based on laboratory extrusion tests and field observations, the face stability of a tunnel driven in cohesive material has been studied by several authors. And recently, more general solution for the tunnel front is given by Leca and Panet(1988). They adopted a limit state design concept to evaluate the face stability of a shallow tunnel driven into cohesionless material and showed that the calculated upper bound solution represented the actual behavior reasonably well. In this study, two factors are simultaneously considered for assessing tunnel face stability: One is the effective stress acting on the tunnel front calculated by upper bound solution; and the other is the seepage force calculated by numerical analysis under the condition of steady state ground water flow. The model tests were performed to evaluate the seepage force acting on the tunnel front and these results were compared with results of numerical analysis. Consequently, the methodology to evaluate the stability of a tunnel face including limit analysis and seepage analysis is suggested under the condition of steady state ground water flow.

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ON THE ORBITAL STABILITY OF INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATIONS WITH SINGULAR POTENTIAL

  • Cho, Yonggeun;Lee, Misung
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1601-1615
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    • 2019
  • We show the existence of ground state and orbital stability of standing waves of nonlinear $Schr{\ddot{o}}dinger$ equations with singular linear potential and essentially mass-subcritical power type nonlinearity. For this purpose we establish the existence of ground state in $H^1$. We do not assume symmetry or monotonicity. We also consider local and global well-posedness of Strichartz solutions of energy-subcritical equations. We improve the range of inhomogeneous coefficient in [5, 12] slightly in 3 dimensions.

Estimation of Ground and Excited State Dipole Moments of Coumarin 450 by Solvatochromic Shift Method

  • Naik, L.R.;Math, N.N.
    • Journal of Photoscience
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    • v.12 no.2
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    • pp.57-61
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    • 2005
  • The ground and excited state dipole moments of Coumarin 450 (C 450) laser dye were measured at room temperature in several solvents of varying dipole moments. The ground state dipole moment (${\mu}_g$) is estimated by using the modified Onsagar model and the excited state dipole moments (${\mu}_e$) were estimated by the method of solvatochromism as well as by utilizing the microscopic solvent polarity parameter ($E^N_T$). Further, the deviation of some of the points from the linearity of the $E^N_T$ versus Stokes shift indicates the existence of specific type of solute-solvent interaction. The excited state dipole moment of C 450 were found to be higher than those of the ground state and is interpreted in terms of the resonance structure of the molecule. A reasonable agreement has been observed between the values obtained by the method of solvatochromism and modified Onsagar model. It is observed that, corresponding to cyclohexane solution, the fluorescence maxima shift towards the red region with increasing the polarity of the solvents, hence the transition involved are of ${\pi}-{\pi}^*$ type.

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GROUND STATE SIGN-CHANGING SOLUTIONS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH INDEFINITE POTENTIALS

  • Yu, Shubin;Zhang, Ziheng
    • Communications of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.1269-1284
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    • 2022
  • This paper is concerned with the following Schrödinger-Poisson system $$\{\begin{array}{lll}-{\Delta}u+V(x)u+K(x){\phi}u=a(x){\mid}u{\mid}^{p-2}u&&\text{ in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=K(x)u^2&&\text{ in }{\mathbb{R}}^3,\end{array}$$ where 4 < p < 6. For the case that K is nonnegative, V and a are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential a is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado [5].

Self-weight Consolidation Analysis of Soft Dredged Clay Ground (준설성토지반의 자중압밀해석)

  • Kim, Hyun-Tae;Lee, Eun-Sung;Kim, Seog-Yeol;Hong, Byung-Man
    • Proceedings of the Korean Society of Agricultural Engineers Conference
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    • 2003.10a
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    • pp.151-154
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    • 2003
  • This paper reviews depositional environments and consolidation characteristic of Soft Dredged Clay fill and then analytical solution of self-weight consolidation is made to find consolidated state. It's known that Soft Dredged Clay Ground is in the under-consolidated state under $U{\fallingdotseq}30%$ from analytical solution. It is effective for higher consolidation rate that the time of Dredge is shorter ani the time of leave is longer. It is conclude that the under-consolidated state should be considered in prediction of consolidation settlement.

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Aerodynamic characteristics of NACA 4412 airfoil section with flap in extreme ground effect

  • Ockfen, Alex E.;Matveev, Konstantin I.
    • International Journal of Naval Architecture and Ocean Engineering
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    • v.1 no.1
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    • pp.1-12
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    • 2009
  • Wing-in-Ground vehicles and aerodynamically assisted boats take advantage of increased lift and reduced drag of wing sections in the ground proximity. At relatively low speeds or heavy payloads of these craft, a flap at the wing trailing-edge can be applied to boost the aerodynamic lift. The influence of a flap on the two-dimensional NACA 4412 airfoil in viscous ground-effect flow is numerically investigated in this study. The computational method consists of a steady-state, incompressible, finite volume method utilizing the Spalart-Allmaras turbulence model. Grid generation and solution of the Navier-Stokes equations are completed using computer program Fluent. The code is validated against published experimental and numerical results of unbounded flow with a flap, as well as ground-effect motion without a flap. Aerodynamic forces are calculated, and the effects of angle of attack, Reynolds number, ground height, and flap deflection are presented for a split and plain flap. Changes in the flow introduced with the flap addition are also discussed. Overall, the use of a flap on wings with small attack angles is found to be beneficial for small flap deflections up to 5% of the chord, where the contribution of lift augmentation exceeds the drag increase, yielding an augmented lift-to-drag ratio.

SHARP THRESHOLDS OF BOSE-EINSTEIN CONDENSATES WITH AN ANGULAR MOMENTUM ROTATIONAL TERM

  • Lu, Zhongxue;Liu, Zuhan
    • Journal of applied mathematics & informatics
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    • v.29 no.3_4
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    • pp.901-908
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    • 2011
  • In this paper, we establish a sharp condition of global existence for the solution of the Gross-Pitaevskii equation with an angular momentum rotational term. This condition is related to the ground state solution of some steady-state nonlinear Schrodinger equation.