QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES

It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical.The potential V is bounded from below and above by positive constants.Because we are considering a critical term which interacts with higher eigenvalues for the linear problem,we need to apply a linking theorem.Notice that the lack of compactness,which comes from critical problems and the fact that we are working in the whole space,are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems.The main feature in this article is to restore some compact results which are essential in variational methods.Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting.This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

SOLUTIONS TO ELLIPTIC SYSTEMS INVOLVING DOUBLY CRITICAL NONLINEARITIES AND HARDY-TYPE POTENTIALS

In this article, an elliptic system is investigated, which involves Hardy-type potentials, critical Sobolev-type nonlinearities, and critical Hardy-Sobolev-type nonlinearities. By a variational global-compactness argument, the Palais-Smale sequences of related approximation problems is analyzed and the existence of infinitely many solutions to the system is established.

SEMICLASSICAL STATES OF HAMILTONIAN SYSTEM OF SCHROEDINGER EQUATIONS WITH SUBCRITICAL AND CRITICAL NONLINEARITIES

We consider the system of perturbed Schroedinger equations{-ε^2△φ＋α（x）φ=β（x）ψ＋Fψ（x,φ,ψ）-ε^2△ψ＋α（x）ψ=β（x）φ＋Fφ（x,φ,ψ）ω：=（φ,ψ）∈H^1（R^N,R^2）where N≥1, α and β are continuous real functions on R^N, and F ： R^N×R^2 → R is of class C^1. We assume that either F（x,ω） is super-quadratic and subcritical in ω∈R^2 or it is of the form ～1/P（x）｜ω｜^p ＋1/2^＊K（x）｜ω｜^2^＊ with p E （2,2^＊） and 2^＊ = 2N/（N-2）, the Sobolev critical exponent, P（x） and K（x） are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution ωε provided ε≤ε; and for any m∈N, there are m pairs of solutions ωε provided that ε≤εm and that F（x, ω） is,in addition, even in ω. Here ε and ωε are sufficiently small positive numbers. Moreover, the energy of ωε tends to 0 as ε→0.

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).

DISCUSSION OF CRITICAL LOAD IN NONLINEAR STABILITY ANALYSIS AND SUGGESTION OF VARIATIONAL PRINCIPLE OF SOLVING CRITICAL LOADS

In this paper, a set of variational formulas of solving nonlinear instability critical loads are established from the viewpoint of variational principle. The paper shows that it is very convenient to solve nonlinear instability critical load by using the variational formulas suggested in this paper.

Influences of aerodynamic loads on hunting stability of high-speed railway vehicles and parameter studies

The influences of steady aerodynamic loads on hunting stability of high-speed railway vehicles were investigated in this study.A mechanism is suggested to explain the change of hunting behavior due to actions of aerodynamic loads：the aerodynamic loads can change the position of vehicle system（consequently the contact relations）,the wheel/rail normal contact forces,the gravitational restoring forces/moments and the creep forces/moments.A mathematical model for hunting stability incorporating such influences was developed.A computer program capable of incorporating the effects of aerodynamic loads based on the model was written,and the critical speeds were calculated using this program.The dependences of linear and nonlinear critical speeds on suspension parameters considering aerodynamic loads were analyzed by using the orthogonal test method,the results were also compared with the situations without aerodynamic loads.It is shown that the most dominant factors a ff ecting linear and nonlinear critical speeds are different whether the aerodynamic loads considered or not.The damping of yaw damper is the most dominant influencing factor for linear critical speeds,while the damping of lateral damper is most dominant for nonlinear ones.When the influences of aerodynamic loads are considered,the linear critical speeds decrease with the rise of cross wind velocity,whereas it is not the case for the nonlinear critical speeds.The variation trends of critical speeds with suspension parameters can be significantly changed by aerodynamic loads.Combined actions of aerodynamic loads and suspension parameters also a ff ect the critical speeds.The effects of such joint action are more obvious for nonlinear critical speeds.

Optimal Neuro-Control Strategy for Nonlinear Systems With Asymmetric Input Constraints

In this paper,we present an optimal neuro-control scheme for continuous-time(CT)nonlinear systems with asymmetric input constraints.Initially,we introduce a discounted cost function for the CT nonlinear systems in order to handle the asymmetric input constraints.Then,we develop a Hamilton-Jacobi-Bellman equation(HJBE),which arises in the discounted cost optimal control problem.To obtain the optimal neurocontroller,we utilize a critic neural network(CNN)to solve the HJBE under the framework of reinforcement learning.The CNN's weight vector is tuned via the gradient descent approach.Based on the Lyapunov method,we prove that uniform ultimate boundedness of the CNN's weight vector and the closed-loop system is guaranteed.Finally,we verify the effectiveness of the present optimal neuro-control strategy through performing simulations of two examples.

Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles

In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0, ∞) ×Ω. For that purpose, following Ibrahim and Majdoub's paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the comparison theorem from Riemannian geometry to estimate the error terms. Finally, using the Strichartz inequality as in Smith and Sogge's paper in 1995, we confirm the global existence.

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.

Pullback Attractors for a Damped Semilinear Wave Equation with Delays

In this paper, we consider the weakly damped wave equations with hereditary effects, and the nonlinearity f satisfies critical growth. The delay term g（t, ut） may be driven by a function with very weak assumptions, namely, just measurability. We analyze the well-posedness of solutions and verify the existence of the pullback ：D-attractor in CH01（Ω） × CL2 （Ω） by constructing the energy functional and combining with the idea of the contractive function.

SOUTHERN BOUNDARY FORCING RELATIVE TO GENESIS,MAINTENANCE AND OSCILLATION OF A SUBTROPICAL HIGH

A nonlinear critical layer and a Kelvin cat's eye excited thereupon are simulated through four schemes in the context of a nonlinear quasi-geostrophic barotropic vorticity equation model with forced stationary wave acting along the southern boundary to investigate effects of tropical steady forcing on the genesis,maintenance and oscillation of a subtropical high(STH).Evidence suggests that the southern forcing is responsible for the planetary quasi-steady anticyclonic Kelvin cat's eye- form flow field inside the nonlinear critical layer,with the eye shifting,vigor and shape changing quite similar to the behaviors of a summer STH,in striking contrast to the northern stationary forcing.As such,the southern boundary-caused cat's eye is likely to be an even more important mechanism for STH genesis and evolution.In addition,a physical mechanism is introduced for quasi-steady planetary wave moving through the critical layer at subtropical latitudes.

ASYMPTOTIC BEHAVIOR OF WAVE EQUATION OF KIRCHHOFF TYPE WITH STRONG DAMPING

The paper deals with the strongly damped nonlinear wave equation of Kirchhoff type. The existence of a global attractor is proven by using the de- composition, and moreover, the structure of the global attractor is established. Our results improve the previous results.