Concentration Variational Method Research Articles

Multiple positive solutions for Kirchhoff-type equations involving critical-concave nonlinearities

In this paper, we research the following Kirchhoff type problem with critical growth (1) 0, & u\\in H^1(\\mathbb{R}^{3}), \\end{cases} \\end{equation}$$]]> { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + u = λ f ( x ) | u | q − 2 u + | u | 4 u , x ∈ R 3 , u > 0 , u ∈ H 1 ( R 3 ) , where 1<q<2, a>0 is a positive constant, 0 $ ]]> b , λ > 0 are real parameters and f ∈ L 6 6 − q ( R 3 ) is a nonzero nonnegative function. We prove the problem has at least two positive solutions by using concentration compactness principle and variational method when b is sufficiently small. Moreover, we also explore the asymptotic behaviour of the positive solutions as b n → n 0 + .

On degenerate fractional Schrödinger–Kirchhoff–Poisson equations with upper critical nonlinearity and electromagnetic fields

This paper intends to study the following degenerate fractional Schrödinger–Kirchhoff–Poisson equations with critical nonlinearity and electromagnetic fields in : where is a positive parameter, , 0<t<1, V is an electric potential satisfying suitable assumptions, and , with , and . With the help of the concentration compactness principle and variational method, and together with some careful analytical skills, we prove the existence and multiplicity of solutions for the above problem as in degenerate cases, that is the Kirchhoff term M can vanish at zero.

Relaxation Time, Excess Relaxation Time and Dipole Moment of Binary Mixture of 2,3-Dichloroaniline and 2-Methoxyethanol using Frequency Domain Reflectometer

Dielectric constant (e') and dielectric loss (e) of 2,3-Dichloroaniline (2,3-DCA), 2-Methoxyethanol (2-ME) and binary mixtures of 2,3-DCA+2-ME in 1,4-dioxane solutions have been measured at microwave frequency 10.985 GHz at different temperatures 20 0 C, 30 0 C, 40 0 C and 50 0 C. Standing microwave techniques and Gopala Krishna’s single frequency concentration variation method have been used for above measurements. The measured values of e¢ and e¢¢ have been used to evaluate dipole moment (µ) and relaxation time (τ). The dielectric relaxation process of binary mixtures containing 70% mole fraction of 2,3-DCA have been calculated. It is found that the dielectric relaxation process can be treated as the rate process like the viscous flow process. Nonlinear variation of relaxation time with molar concentration of 2,3-DCA in the whole concentration range of the binary mixture indicates the existence of solute-solute and solute–solvent type of molecular association was predicted. Dipole moment (µ) of 2,3-DCA and 2-ME slightly non-linearly increases with rise in temperature. Excess inverse relaxation time (1/τ) e calculated at different temperatures are found to be positive. All parameters (e'), (e), (τ), (1/τ) e and (μ) are calculated and discussed to yield information on the dipole alignment and molecular rotation of the binary liquid mixtures. From all the derived dielectric parameters, molecular interactions are interpreted through hydrogen bonding. DOI: http://dx.doi.org/10.17807/orbital.v12i2.1368

Effect of temperature on the dielectric relaxation time of binary mixture of 2-chloroaniline and 2-ethoxyethanol in 1,4-Dioxane solution using microwave absorption data

The dielectric constant ( ),dielectric loss ( ), relaxation times (τ) and the dipole moments (µ) of the binary mixtures of different molar concentrations of 2-chloroaniline C6H6ClN in the binary mixtures of 2-chloroaniline (2-CA) and 2-ethoxyethanol (2-EE) C4H10O2 in 1,4-dioxane C4H10O2 solutions were calculated by using standard standing wave microwave X- band techniques and Gopala Krishna’s single frequency 10.985 GHz concentration variational method at different temperatures 20,30,40,50 0 C. The dielectric relaxation process was found to be an activated process. The dielectric relaxation process of binary mixtures containing 30% mole-fraction of 2-CA were calculated at the respective given temperatures. On the basis of the observations, it was found that the dielectric relaxation process could be treated as the rate process like the viscous flow process. The dipole moment of pure and binary mixtures were calculated with varying temperatures. The solute-solute and solute-solvent molecular associations were predicted. Dipole moment (µ) of 2-CA and 2-EE non- linearly increased with rise in temperature.

Dielectric relaxation study of binary mixture of sulfolane and N,N-dimethylformamide in benzene solution using microwave absorption data

The dielectric constant (ɛ′) and dielectric loss (ɛ″) for dilute solutions of the binary mixture of different molar concentrations of sulfolane and DMF in benzene solution has been measured at 9.885 GHz and different temperatures (25, 30, 35, 40°C) by using standard microwave techniques. Following the single frequency concentration variational method, the dielectric relaxation time (τ) and dipole moment (μ) have been calculated. It is found that dielectric relaxation process can be treated as the rate process, just like the viscous flow. The presence of solute-solute molecular associations in benzene solution has been proposed. Energy parameters (ΔHɛ, ΔFɛ, ΔSɛ) for dielectric relaxation process of binary mixture at 50% mole fraction in benzene at 25, 30, 35, and 40°C have been calculated and compared with the corresponding energy parameters (ΔHη, ΔFη, ΔSη) for the viscous flow.

Dielectric relaxation study of binary mixture of sulfolane and N,N-dimethylacetamide in benzene solution using microwave absorption data

The dielectric constant (\( \varepsilon ^{\prime} \)) and dielectric loss (\( \varepsilon ^{\prime \prime} \)) for dilute solutions of the binary mixture of different molar concentrations of sulfolane and DMA in benzene solution have been measured at 9.885 GHz at different temperatures by using standard microwave techniques. Following the single frequency concentration variational method, the dielectric relaxation time (\( \tau \)) and dipole moment (\( \mu \)) have been calculated. It is found that dielectric relaxation process can be treated as the rate process, just like the viscous flow. The presence of solute–solute molecular associations in benzene solution has been proposed. Energy parameters (\( \Updelta H_{\varepsilon } ,\Updelta F_{\varepsilon } ,\Updelta S_{\varepsilon } \)) for dielectric relaxation process of binary mixture at 50 % mole fraction in benzene at different temperatures have been calculated and compared with the corresponding energy parameters (\( \Updelta H_{\eta } ,\Updelta F_{\eta } ,\Updelta S_{\eta } \)) for the viscous flow.

Dielectric relaxation studies of binary mixtures of tetrahydrofuran and N,N-dimethylformamide in benzene solutions using microwave absorption data

The dielectric relaxation times $$ \left( \tau \right) $$ and dipole moments $$ \left( \mu \right) $$ of the binary mixtures of different molar concentrations of tetrahydrofuran (THF) and N,N-dimethylformamide (DMF) in benzene solution have been calculated by using standard standing wave microwave techniques and Gopala Krishna’s single frequency(9.90 GHz) concentration variational method at different temperatures (25, 30, 35, and 40 °C). The energy parameters (∆H e , ∆F e , ∆S e ) for the dielectric relaxation process of THF + DMF binary mixture containing 50 mol% THF have been calculated at different temperatures (25, 30, 35, and 40 °C) and comparison has been made with the corresponding energy parameters (ΔH η , ΔF η , ΔS η ) for viscous flow process. It is found that dielectric relaxation process can be treated as rate process just like the viscous flow process. On the basis of above studies, solute–solute and solute–solvent types of molecular associations have been predicted.

Dielectric relaxation study of sulfolane in carbon tetrachloride solution from microwave absorption data

The dielectric constant (ɛ′) and dielectric loss (ɛ″) for dilute solutions of sulfolane in carbon tetrachloride solution have been measured at 9.885 GHz at different temperatures viz. 25°C, 30°C, 35°C and 40°C by using standard microwave techniques. Following the single frequency concentration variational method, the dielectric relaxation time (τ) and dipole moment (µ) have been calculated. It is found that dielectric relaxation process can be treated as the rate process, just like the viscous flow. Based on the above studies, monomer structure of sulfolane in carbon tetrachloride has been inferred. Energy parameters (ΔHg, ΔFg, ΔSg) for dielectric relaxation process of sulfolane in carbon tetrachloride at 25°C, 30°C, 35°C and 40°C have been calculated and compared with the corresponding energy parameters (ΔHη, ΔFη, ΔSη) for the viscous flow.

Temperature Dependence Studies on Dielectric Relaxation and Electric Dipole Moment of Poly (Ethylene Glycol)

Temperature dependant studies of dielectric relaxation time τ and electric dipole moment μ of poly (ethylene glycol) (PEG) with molecular weight 400 in dilute solution in benzene was carried out at 9.59 GHz at different temperatures using an X-band microwave bench. GopalaKrishna's single frequency and concentration variation method were adopted to determine τ and μ. Results are reported and discussed. On the basis of these parameters, the dielectric behavior of PEG-400 in a nonpolar solvent is discussed.

Dielectric relaxation of sulfolane in benzene solution from microwave absorption data

The electric constant (ɛ′) and dielectric loss (ɛ″) for dilute solutions of sulfolane in benzene solution has been measured at 9.885 GHz at different temperatures (25, 30, 35, and 40°C) by using standard microwave techniques. Following the single frequency concentration variational method, the dielectric relaxation time (τ) and dipole moment (μ) have been calculated. It is found that dielectric relaxation process can be treated as the rate process, just like the viscous flow. Based on the above studies, monomer structure of sulfolane in benzene has been inferred. The presence of solute-solvent associations in benzene solution has been proposed. Energy parameters (ΔHɛ, ΔFɛ, ΔSɛ) for dielectric relaxation process of sulfolane in benzene at 25, 30, 35, and 40°C have been calculated and compared with the corresponding energy parameters (ΔHη, ΔFη, ΔSη) for the viscous flow.

Molecular Association of Tetramethylurea and Chlorobenzene Molecules in Microwave Frequency Range

The dielectric constant ε´ and dielectric loss ε´´ of the binary mixtures of tetramethylurea (TMU) and chlorobenzene (CB) have been calculated at 9.883 GHz by using standard standing microwave techniques. Gopalakrishna’s single frequency concentration variation method has been used to calculate dipole moment μ and dielectric relaxation time τ for different mole fractions of TMU in the binary mixture at different temperatures of 25 °C, 30 °C, 35 °C, and 40 °C. The variation of dielectric relaxation time with the mole fraction of TMU in the whole concentration range of the binary mixtures was found to be non-monotonic. The solute-solute and solute-solvent type of molecular associations may be proposed based upon above observations. Using Eyring rate equations the energy parameters ΔH, ΔF, and ΔS for the dielectric relaxation process and the viscous flow process have been calculated at the given temperatures. It is found from the comparison of energy parameters that, just like the viscous flow process, the dielectric relaxation process can also be treated as a rate process.

MOLECULAR INTERACTION STUDY OF N-BENZYLFORMAMIDE WITH 1-ALCOHOLS USING X-BAND MICROWAVE

The secondary amide unit is a subject of particular interest, because of its occurrence in peptides and proteins. Molecular interaction between N-benzylformamide (NBF) with 1-alcohols (1-propanol, 1-butanol, 1-pentanol) has been studied in carbon tetrachloride by using X-band microwave bench at 936 GHz. Dielectric constant (ε') and dielectric loss (ε″) of alcohol and NBF and their binary mixture for different mole fractions of NBF have been determined. Dielectric relaxation time (τ) of the binary system is obtained by both Higasi's method and the Gopalakrishna single-frequency concentration variational method. The results show that the most likely interaction between alcohols and NBF is 1:1 complex for binary mixture through the free hydroxyl group of the alcohol and the carbonyl group of NBF. The alkyl chain-length of both alcohol and amide plays an important role in the determination of the strength of hydrogen bond ( O – H : C = O ) formed. The variation of relaxation time of NBF+1-alcohol mixtures in CCl4indicates a weak solute-solvent type of molecular association. The result shows that as the relaxation time of the proton acceptor increases, the donating ability of the solute environment increase.

Dielectric Relaxation of Binary Mixtures of Tetrahydrofuran and N-Methylformamide in Benzene Solution Using Microwave Absorption Studies

Using standard standing wave microwave X-band technique and following Gopala Krishna’s single frequency (9.90 GHz) concentration variational method, the dielectric relaxation times τ and the dipole moments μ of dilute solutions of Tetrahydrofuran (THF), N-methylformamide (NMF), and THF+NMF binary mixtures in benzene solutions have been calculated at different temperatures (25°C, 30°C, 35°C, and 40°C). The energy parameters (ΔHε , ΔFε , ΔSε ) for the dielectric relaxation process for the THF+NMF binary mixture containing 30 mol% THF have been calculated at 25°C, 30°C, 35°C, and 40°C and compared with the corresponding viscosity parameters. A good agreement between the free energy of activation from these two sets of values shows that the dielectric relaxation process like the viscous flow process can be treated as a rate process. From relaxation time behaviour of THF and NMF binary mixture in benzene solution, solute-solute types of the molecular association has been proposed.

Dielectric studies on some mesogenic, non-mesogenic and organic molecules

Two important molecular parameters, namely, the dielectric relaxation time τ and electric dipole moment are highly useful in having insight into the molecular structure, size, shape, apart from the inter- and intra-molecular forces, etc. With these in view dielectric measurements in benzene at room temperature on the pure samples of o-ethyl phenol, 2-n-butylphenol, 4-n-butylphenol, 2,6-dimethoxyphenol and 3,4-difluorophenol were carried out at a frequency of 9.98 GHz by employing concentration variation method. Similar measurements, on a single weight fraction of each of them at 9.98 GHz and also at 8.74 GHz are carried out. Measurements on a single weight fraction in benzene of each of the liquid crystal samples, namely, EPCP.car (Butyl-p-(p-ethoxy phenoxy corbonyl)-phenyl carbonate), PPPB (p-pentylphenyl-p-propyl benzonate), EPAP.Hp (p-(p-ethoxyphenyl azo) phenyl heptenate) and EPAP. Und (p-(p-ethoxyphenyl azo) phenyl undecylenate were also carried out at the said two frequencies. Using the obtained values of ε′, ε′′ relaxation time and dipole moment were determined using different methods. The obtained results are compared with earlier results wherever available.

Dielectric Relaxation of Benzonitrile in Benzene

The dielectric constant, ε′ , and dielectric loss, ε″ , of dilute solutions of benzonitrile (C6H5CN) in benzene have been measured at 9.885 GHz at 25, 30, 35, and 40 °C using standard standing microwave techniques. Following the single frequency concentration variational method of Gopala Krishna, the dielectric relaxation time, τ , and the dipole moment, μ, at various temperatures have been calculated. It was concluded that dielectric relaxation processes can be treated as rate processes just like the viscous flow process. Based on the above studies, the monomer structure of benzonitrile in benzene has been inferred. The solute-solvent molecular association of benzonitrile in benzene has been found. The energy parameters ΔHε , ΔFε , ΔSε for the dielectric relaxation process of benzonitrile in benzene at different temperatures have been calculated and compared with the corresponding energy parameters ΔHη , ΔFη , ΔSη for the viscous flow process.

Dielectric Relaxation Studies of Binary Mixtures of N-Methylformamide and Tetramethylurea in Benzene Using Microwave Absorption Data

The dielectric relaxation time (τ ) and dipole moment (μ) of binary mixtures of different molar concentrations of N-methylformamide (NMF) in binary mixtures of NMF and tetramethylurea (TMU) in benzene have been calculated at 25, 30, 35 and 40 ◦C using standard standing wave microwave techniques and following the single frequency (9.885 GHz) concentration variational method of Gopala Krishna. The energy parameters (ΔH ε , ΔF ε , ΔS ε ) for the dielectric relaxation process of binary mixtures containing 30 mol% of NMF have been calculated at different temperatures, and comparison has been made with the corresponding energy parameters (ΔH η , ΔF η , ΔS η ) for the viscous flow process. Based on these studies, it was inferred that the dielectric relaxation process can be treated as a rate process just like the viscous flow process. Solute-solvent and solute-solute molecular associations have been proposed.

Dielectric Relaxation Studies of Binary Mixtures of Ethanol and Chlorobenzene in Benzene Solution from Microwave Absorption Data

Dielectric relaxation of different molar concentrations of ethanol (C2H5OH) in binary mixtures of ethanol and chlorobenzene in benzene solutions has been studied at 9.883 GHz by using standard standing microwave techniques and Gopala Krishna’s single frequency concentration variation method at different temperatures (25 °C, 30 °C, 35 °C, and 40 °C). It was found that the dielectric relaxation time varies linearly with variation in the molar concentration of ethanol in the whole concentration range of the binary mixture. Based upon these results the absence of solute-solute and the presence of solute-solvent type of molecular associations could be assumed. The energy parameters for the dielectric relaxation process of binary mixtures containing 50 mol% of ethanol have been calculated at the given temperatures. Comparison has been made with the corresponding energy parameters for viscous flow processes. It was found that the dielectric relaxation process can be treated as the rate process like the viscous flow process.

Dielectric relaxation studies of binary mixtures of N-methylacetamide and N-methylformamide in benzene solutions using microwave absorption data

Using standard microwave X-band technique and by following Gopala Krishna's single frequency (9.90 GHz) concentration variational method, the dielectric relaxation times (τ) and the dipole moments (μ) of dilute solution of N-methylacetamide (NMA), N-methylformamide (NMF) and NMA + NMF binary mixtures in benzene solutions have been calculated at different temperatures. The energy parameters for the dielectric relaxation process for NMA + NMF binary mixture containing 30 mol% NMF have been calculated at 25, 30, 35 and 40°C and compared with the corresponding viscosity parameters. A good agreement between the free energy of activation from these two sets of values shows that the dielectric relaxation process like the viscous flow process can be treated as the rate process. From relaxation time behavior of NMA and NMF binary mixture in benzene solution, solute–solute types of the molecular association has been proposed.

Dielectric Relaxation Study of Ethanol in Benzene from Microwave Absorption Data

The dielectric constant (ε’) and dielectric loss (ε″) of dilute solutions of ethanol in benzene have been measured at 9.883 GHz at 25, 30, 35 and 40 °C using standard microwave techniques. The dielectric relaxation time (τ) and dipole moment (μ) at 25, 30, 35 and 40 °C have been calculated by using the single frequency concentration variation method suggested by Gopala Krishna. It is found that the dielectric relaxation process can be treated as a rate process like the viscous flow process. A monomer structure of C2H5OH in benzene solution has been inferred. Based upon these studies, the presence of solute-solvent associations has been proposed. The energy parameters for the dielectric relaxation process have been calculated and compared with the corresponding energy parameters of viscous flow.

Dielectric Relaxation Studies of Mixtures of N-Methylacetamide and Ethanol in Benzene Solutions Using Microwave Absorption Technique

Using standard standing wave microwave X-band techniques, and by following Gopala Krishna’s single frequency (9.90 GHz) concentration variational method, the dielectric relaxation times (τ) and dipole moments (μ) of binary mixtures of different molar concentrations of ethanol (EtOH) in binary mixtures of N-methylacetamide (NMA) and ethanol in benzene solutions at 25, 30, 35 and 40 ◦C have been calculated. The activation parameters (ΔHε , ΔFε , ΔSε ) for the dielectric relaxation process of binary mixtures containing 30 mol% of EtOH have been calculated at 25, 30, 35 and 40 ◦C and compared with the corresponding viscosity parameters. A good agreement between the free energy of activation from these two sets of values shows that the dielectric relaxation process, like the viscous flow, can be treated as a rate process. From relaxation time behaviour of NMA and EtOH binary mixtures in benzene solution, solute-solute and solute-solvent types of the molecular association have been predicted.