On small perturbations of the Schrödinger operator with a periodic potential

Consideration is given to small perturbations of a potential, periodic with respect to the variables x_j, j=1, 2, 3, by a function periodic in x₁ and x₂ that decays exponentially as |x₃| → ∞. It is shown that in the neighborhood of energies corresponding to the extrema in the third quasimomentum component of nondegenerate eigenvalues of the Schrödinger operator, with the periodic potential considered in a cell, there exists a unique solution (up to within a numerical factor) to the integral equation describing both the eigenvalues and resonance levels. The asymptotic behavior of the eigenvalues and resonance levels is investigated.     

On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.     

On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth

In this paper we study the existence of solutions for fractional Schrödinger equations of the form where V is a potential bounded and the nonlinear term has the critical exponential growth. We prove the existence of at least one weak solution by combining the mountain‐pass theorem with the Trudinger–Moser inequality and a version of a result due to Lions for critical growth in .     

On singular perturbations of a Steklov-type problem with asymptotically degenerate spectrum

Doklady Mathematics -     

On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth

We consider the equation −Δu+V(x)u=f(x,u) for $\text{x∈}\text{R}{}^2$ where $\text{V:}\text{R}{}^2\text{→}\text{R}$ is a positive potential bounded away from zero, and the nonlinearity $\text{f:}\text{R}{}^2\text{×}\text{R}\text{→}\text{R}$ behaves like exp(α|u|²) as |u|→∞. We also assume that the potential V(x) and the nonlinearity f(x,u) are asymptotically periodic at infinity. We prove the existence of at least one weak positive solution $\text{u∈H}{}^1\text{(}\text{R}{}^2\text{)}$ by combining the mountain-pass theorem with Trudinger–Moser inequality and a version of a result due to Lions for critical growth in $\text{R}{}^2$.     

On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system:

where A is a real 2×2 constant matrix, and f(t,0,)=O() and ∂_xf(t,0,)=O() as →0. With some non-resonant conditions of the frequencies with the eigenvalues of A and without any nondegeneracy condition with respect to , by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter .     

Global dynamics of the periodic logistic system with periodic impulsive perturbations

This paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations. The results of D.D. Bainov and P.S. Simeonov (1993) are extended and dynamics different from the corresponding continuous system are found. It is shown that the system may have a unique positive periodic solution which is globally asymptotically stable, or go extinct when the two periods are rational dependent. When they are rational independent, the system has no periodic solutions, however, still has a global attractor or go extinct under some conditions.     

具有线性脉冲的周期捕食系统的持久性

研究具有HollingIV功能性反应和脉冲的周期捕食食饵系统．找到了影响该系统动力学行为的阈值Ro．证明了当Ro〈1时,该系统的食饵灭绝周期解是局部渐近稳定的;当R0〉1时,该系统的食饵灭绝周期解变得不稳定且食饵将一致持久．     

On the solutions of a second order nonlinear system with almost periodic forcing

A second order almost periodic perturbed nonlinear system with small parameter is discussed in this paper. Based on some analytical techniques and the method of averaging, some sufficient conditions are obtained for existence of one almost periodic solution. Also some suitable conditions are given for existence of two almost periodic solutions. The obtained new results generalized the known results in Seifert [1], [3] and He [2], [4]. Finally, some applications are presented to illustrate that our results are a good generalization of the known results.     

Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources

This paper deals with the global existence and blow-up of nonnegative solution of the degenerate reaction-diffusion system with nonlinear localized sources involved a product with local terms. We investigate the influence of localized sources and local terms on global existence and blow up for this system. Moreover, we establish the precise blow-up estimates. Finally, for the special case p₁=p₂=0, we show the blow-up set is whole region and the uniform blow-up profiles are obtained. These extend a resent work of Chen and Xie in [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004) 79-93], which considered the single equation with localized sources.     

Stability of plane Couette flows with respect to small periodic perturbations

We consider the plane Couette flow v₀=(x_n,0,…,0) in the infinite layer domain , where n≥2 is an integer. The exponential stability of v₀ in Lⁿ is shown under the condition that the initial perturbation is periodic in (x₁,…,x_n−1) and sufficiently small in the Lⁿ-norm.     

On a degenerate parabolic system

In this paper we study a degenerate parabolic system

     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.