一类临界拟线性椭圆型方程组解的存在性

本文讨论了一类临界拟线性椭圆型方程组解的存在性问题.利用LionsPL提出的第二集中紧性原理和山路引理,证明了该方程组在超线性扰动情形下非平凡解的存在性.此外,利用极值原理,也得到了该方程组在次线性扰动情形存在非平凡解.     

非对称p-Laplacian Dirichlet问题的非平凡解

研究了一类非对称的p-Laplacian(p1)Dirichlet问题.在正半轴不需要假设Ambrosetti-Rabinowitz的超二次条件下,利用山路定理建立非平凡解的存在性结果.     

一类高阶线性微分方程解的复振荡

研究了齐次微分方程f^(k) bf1 ezf=0的复振荡问题,其中b为复常数,在假设了方程存在非平凡解且其零点的密指量等价于o(e^r)的条件下,得到了方程的非平凡解f的一般表达式。     

一类双相问题多重解的存在性（英文）

本文研究一类双相问题多重解的存在性.基于变分方法,证明了该问题至少存在两个非平凡解.当非线性项关于u是奇函数,利用对称山路引理,我们同时得到了该问题存在无穷多对解.     

源自人口动力学的半线性p-Laplace的Dirichlet问题解

研究源自人口动力学的半线性p-Laplace方程的Dirichlet问题,得到了该问题在零点处的能量泛函是平凡的Morse临界群.因而,确定了该问题非平凡解的存在性及其分岔性.     

关于一类带有非线性边界条件的拟线性椭圆方程

本文研究了一类带有非线性边界条件的拟线性椭圆方程.在比常用的经典条件弱的假设条件下,得到该方程所对应的能量泛函存在有界Palais-Smale序列.然后,利用山路引理,得到该方程非平凡解的存在性.最后,给出一个例子,说明所给的条件比经典条件弱.     

一类椭圆型变分不等式的非平凡广义解的存在性

本文研究了下列障碍问题的非平凡解的存在性u∈K∶∫Ωu.(u-u)dx ∫Ωa(x)u.(v-u)dx∫Ωp(x,u)(v-u)dx,v∈K.其中K={v∈H01(Ω)∶vψa.e.onΩ}.利用关于不等式推广的山路引理,在a(x)和障碍p(x,ξ)满足适当的假设下,我们证明了上述不等式存在非平凡解.     

一类带退化椭圆算子的非局部方程解的存在性

利用临界点理论和分析技巧,证明一类带退化椭圆算子的非局部方程在适当的假设条件下非平凡解的存在性,所得结论丰富和发展了已有文献的相关结果.     

分数阶耦合非线性Schrdinger方程组的山路解

该文研究一类非线性分数阶Schrdinger方程组Dirichlet问题非平凡解的存在性.所用主要工具是分数阶Sobolev空间上的山路引理.要点是证明PS条件及该方程组的山路解是非平凡的.     

一类椭圆型半变分不等式解的存在性(英文)

本文研究了一类Dirichlet边界的椭圆型半变分不等式问题.利用非光滑形式的环绕定理和非光滑形式的对称山路定理,得到了在相应假设条件下此不等式问题至少有一个非平凡解和无穷多解.本文中非光滑势能在原点处关于算子+V(x)的第一正特征值λ是不完全共振的.     

临界群与二阶差分方程解的多重性

研究了一类二阶非线性差分方程两点边值问题解的多重性.当该问题的非线性项在无穷远点具有特殊的渐近线性性质时,利用变分方法,结合临界群与Morse理论,同时考虑正、负能量泛函的临界点,不论该问题是否发生共振,均证明了它至少存在两个非零解.     

Neumann problems resonant at zero and infinity

We consider a semilinear Neumann problem with a reaction which is resonant at both zero and ±∞. Using a combination of methods from critical point theory, together with truncation techniques, the use of upper–lower solutions and of the Morse theory (critical groups), we show that the problem has at least five nontrivial smooth solutions, four of which have constant sign (two positive and two negative).     

The Stability of a Flexible Vertical Rod on a Vibrating Support

The classical Kapitsa problem of the inverted flexible pendulum is generalized. We consider a thin homogeneous vertical rod with a free top end and pivoted or rigid attached lower end under the weight of the pendulum’s action and vertical harmonic vibrations of the support. In both cases of attachment, we have stability conditions for the vertical rod position. We take the influence of axial and bending rod vibrations and describe the bending vibrations using the Bernoulli–Euler beam model. The solution is built as a Fourier expansion by eigenfunctions of auxiliary boundary-value problems. As a result, the problem is reduced to the set of ordinary differential equations with periodic coefficients and a small parameter. The asymptotic method of two-scale expansions is used for its solution and to determine the critical level of vibration. The influence of longitudinal waves in the rod essentially decreases the critical load. The single-mode approximation has an acceptable accuracy. With pivoting support at the lower end of the rod, we find the explicit approximate solution. For the rigid attachment, we conduct numerical analysis of the critical level of vibrations depending on the problem parameters.     

Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange

We compare Euler-Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation.     

On elliptic problems with two critical Hardy–Sobolev exponents at the same pole

In this paper we study an elliptic problem involving two different critical Hardy–Sobolev exponents at the same pole. By variational methods and concentration compactness principle, we obtain the existence of positive solution to the considered problem.     

On the second solution to a critical growth Robin problem

We investigate the existence of the second mountain-pass solution to a Robin problem, where the equation is at critical growth and depends on a positive parameter λ. More precisely, we determine existence and nonexistence regions for this type of solutions, depending both on λ and on the parameter in the boundary conditions.     

Error estimates for the Ginzburg-Landau approximation

Modulation equations play an essential role in the understanding of complicated dynamical systems near the threshold of instability. Here we look at systems defined over domains with one unbounded direction and show that the Ginzburg-Landau equation dominates the dynamics of the full problem, locally, at least over a long time-scale. As an application of our approximation theorem we look here at Bénard's problem. The method we use involves a careful handling of critical modes in the Fourier-transformed problem and an estimate of Gronwall's type.     

Multiple solutions for inhomogeneous critical semilinear elliptic problems

In this paper, we consider the existence of multiple positive solutions for an inhomogeneous critical semilinear elliptic problem. We show that the problem possesses at least four positive solutions.     

On the discrete boundary value problem for anisotropic equation

In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions.     

On the No-Gap Second-Order Optimality Conditions for a Discrete Optimal Control Problem with Mixed Constraints

Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions.