MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRODINGER EQUATIONS

In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^(N),(0.1)where N≥4,2≤p<2^(*),2_α^(*)=(2N-α)/(N-2)with 0<α<4,λ>0,μ∈R,A(x)=(A_(1)(x),A_(2)(x),…,A_(N)(x))is a real local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on R^(N).Supposing thatΩ=int V^(-1)(0)■R^(N)is bounded,we show that problem(0.1)possesses at least cat_(Ω)(Ω)nontrivial solutions ifλis large.

Experimental research on the critical conditions and critical equation of chip splitting when turning a C45E4 disc workpiece symmetrically with a high-speed steel double-edged turning tool

Chip splitting is a natural chip separation phenomenon that can significantly reduce cutting energy consumption.To reveal its occurrence mechanisms,a method for obtaining its critical conditions through cutting experiments and establishing its critical equation is proposed in this paper.Based on previous research results regarding the relationship between chip removal interference and chip splitting,the control variables that affect chip splitting are identified by analyzing a geometric model of the cutting process.A total of 366 experiments on turning a C45E4 disc workpiece with a high-speed steel double-edged turning tool based on the dichotomy method were conducted and 51 experimental data on chip splitting critical conditions were obtained.Accordingto these experimental data,a critical equation expressed by a finitedegree polynomial with a cutting thickness equal to the other control variables was fitted.By analyzing the critical surface,it was determined that chip splitting followed a law in which the smaller the cutting thickness and the larger the absolute value of the negative rake angle,edge angle,and edge inclination of the tool,the more likely chip splitting was to occur.Through a verification experiment,it was determined that the derived critical equation could accurately predict the occurrence of 95.24%of chip splitting.It was also determined that the occurrence of chip splitting led to a cliff-like drop in the specific total cutting force with a maximum drop of 51.23%.This research lays a foundation for the rational utilization of chip splitting in tool structure parameter design and cutting parameter energy saving optimization.

EIGENFUNCTIONS OF THE NONLINEAR ELLIPTIC EQUATION WITH CRITICAL EXPONENT IN R^2

We consider the following eigenvalue problem: [GRAPHICS] Where f(x, t) is a continuous function with critical growth. We prove the existence of nontrivial solutions.

INFINITELY MANY SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

Entire Sign-Changing Solutions to the Fractional Critical Schrodinger Equation

In this paper,we consider the fractional critical Schrödinger equation(FCSE)(-Δ)^(s)u-|u|2^(*)s-2 u=0,where u∈˙H^(s)(R^(N)),N≥4,0<s<1 and 2^(*)s=2 N/N-2 s is the critical Sobolev exponent of order s.By virtue of the variational method and the concentration compactness principle with the equivariant group action,we obtain some new type of nonradial,sign-changing solutions of(FCSE)in the energy space˙H^(s)(R^(N)).The key component is that we take the equivariant group action to construct several subspace of˙H^(s)(R^(N))with trivial intersection,then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of(FCSE)in˙H^(s)(R^(N)).

Model of critical strain for dynamic recrystallization in 10%TiC/Cu-Al_2O_3 composite

Using the Gleeble-1500 D simulator, the hot deformation behavior and dynamic recrystallization critical conditions of the 10%Ti C/Cu-Al2O3(volume fraction) composite were investigated by compression tests at the temperatures from 450 °C to 850 °C with the strain rates from 0.001 s-1 to 1 s-1. The results show that the softening mechanism of the dynamic recrystallization is a feature of high-temperature flow true stress-strain curves of the composite, and the peak stress increases with the decreasing deformation temperature or the increasing strain rate. The thermal deformation activation energy was calculated as 170.732 k J/mol and the constitutive equation was established. The inflection point in the lnθ-ε curve appears and the minimum value of-(lnθ)/ε-ε curve is presented when the critical state is attained for this composite. The critical strain increases with the increasing strain rate or the decreasing deformation temperature. There is linear relationship between critical strain and peak strain, i.e., εc=0.572εp. The predicting model of critical strain is described by the function of εc=1.062×10-2Z0.0826.

The Existence of a Weak Solution of Inhomogeneous Quasilinear Elliptic Equation with Critical Growth Conditions

In this paper, we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions: where N≥2, f（x,u）～|u|m-1eb|u|γat +∞, with γ=N/N-1, m≥1, b】0.

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem.The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem.Compared to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).

Periodic Solutions to a Kind of Second Order Neutral Functional Differential Equation in the Critical Case

In this paper, the authors consider the existence of periodic solutions for a kind of second neutral functional differential equation as follows：（x（t） - cx（t -τ）＂ = g（t, x（t - μ（t））） ＋ e（t）,in the critical case ｜c｜ = 1. By employing Mawhin＇s continuation theorem and some analysis techniques, some new results are obtained.

Growth Characteristics and Resource Evaluation of Sebastes schlegelii in Zhangzidao Artificial Reef Area

[Objective]The paper was to formulate catchable size and total allowable catch of Sebastes schlegelii in Zhangzidao artificial reef area.[Method]Based on analysis of length-weight formula,body length and weight growth equations,and instantaneous mortality rate,the inflection age and critical age of weight growth were calculated,and the biomass of S.schlegelii in Zhangzidao artificial reef area was estimated.[Result]The growth equation of body length was Lt=412.5×[1-e^-0.21（t＋0.65）]and the growth equation of body weight was Wt=1 734.2×[1-e^-0.21（t＋0.65）]^2.92.The inflection age and critical age for weight growth of S.schlegelii were 4.45 and 4.82 a,respectively.The biomass in Zhangzidao artificial reef area was about 456.8 t.[Conclusion]For S.schlegelii flock in Zhangzidao artificial reef area,the catchable length was about 271.2-281.7 mm,the catchable weight as about 509.4-569.5 g,and the total allowable catch was about 60.43 t.

Influence of the gassing materials on the dielectric properties of air

Influence of the gassing materials, such as PA6, PMMA, and POM on the dielectric properties of air are investigated. In this work, the fundamental electron collision cross section data were carefully selected and validated. Then the species compositions of the air–organic vapor mixtures were calculated based on the Gibbs free energy minimization. Finally, the Townsend ionization coefficient, the Townsend electron attachment coefficient and the critical reduced electric field strength were derived from the calculated electron energy distribution function by solving the Boltzmann transport equation. The calculation results indicated that H;O with large attachment cross sections has a great impact on the critical reduced electric field strength of the air–organic vapor mixtures. On the other hand, the vaporization of gassing materials can help to increase the dielectric properties of air circuit breakers to some degree.

Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.