全空间中一类非局部问题非平凡解的存在性

利用变分方法研究了R~N上一类带有临界非线性项的p-Kirchhoff型问题非平凡解的存在性.首先得到了该问题的能量泛函并证明了其具有山路引理的几何结构.其次给出了山路值c的一个上界并且证明了相应的(PS)_c序列是有界的.最终利用集中紧性原理及其它相关知识证明了能量泛函满足(PS)_c条件,从而表明了能量泛函存在非零的临界点,即证明了该问题至少存在一个非平凡解.     

R~N上一类临界p-Kirchhoff型问题非平凡解的存在性

主要通过变分方法研究了R~N上一类带有临界非线性项的p-Kirchhoff型问题非平凡解的存在性.首先得到了问题的能量泛函并证明了其具有山路引理的几何结构,由此获得了能量泛函的一个(PS)_c序列.其次证明了此(PS)_c序列有界并且给出了c的一个上界.最终利用相关知识证明了此(PS)_c序列存在收敛子列,从而证明了问题至少存在一个非平凡解.     

含有对数非线性项的p-Laplace方程的多重解

主要通过变分方法研究了有界区域上含有变号权函数和对数非线性项的一类p-Laplace方程Dirichlet边值问题的多解性.通过分解能量泛函的Nehari流形,利用对数Sobolev不等式,极小化序列方法及相关知识证明了能量泛函至少存在两个非零极小元,从而证明了问题至少存在两个非平凡解.     

非线性弹性理论的泛变分原理

本文从泛能量泛函[注]出发,提出并证明了非线性弹性理论静力学(或动力学)的非保守、有跳变和分区问题的统一变分原理——泛变分原理[注]。由泛能量待定泛函出发可直接得出各种变分原理。     

一类奇异临界椭圆方程非平凡解的存在性

研究了一类带Sobolev-Hardy临界指数的奇异椭圆方程,应用变分方法,通过能量估计和证明对应的能量泛函满足(PS)_c条件,运用山路引理得到了这类方程非平凡解的存在性.     

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.     

含临界位势的广义平均曲率方程的解

讨论一个含临界位势的广义平均曲率方程在Dirichlet边界条件下解的存在性.此方程相应的变分泛函关于u的梯度非齐次,且Sobolev空间嵌入失去紧性.为了克服这些困难,本文将关于范数的一个基本结论推广到一般的偶泛函,并利用C.K.N不等式及Ambrosetti的山路引理证明了方程存在非平凡解.     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

大变形弹塑性率变分极值原理

在Updated Lagrangian率形式下,研究了大变形弹塑性率问题的对偶极值变分原理.证明了变分泛函的凸性取决于一个所谓的间隙函数.     

含极限次临界增长项p-Laplace方程的无穷多解

讨论了有界光滑区域上一类p-Laplace方程,非线性项具奇对称性且在无穷远为极限次临界增长．证明了变分泛函在大范围内满足推广的Palais-Smale条件,构造了变分泛函的一列临界值,进而得到了无穷多个弱解的存在性,对应泛函的能量趋于正无穷．所得到的结果推广了次临界增长的情形．     

Existence of solutions for discrete p-Laplacian with potential boundary conditions

We are concerned with the existence of solutions for some discrete p-Laplacian equations subjected to a potential type boundary condition. Our approach is a variational one and relies on Szulkin's critical point theory. We obtain the existence of solutions in a coercive case as well as the existence of non-trivial solutions when the corresponding energy functional has a ‘mountain pass’ geometry.     

Mountain Pass Solutions for a Double-Well Energy

We establish the existence of a mountain pass solution for a variational integral involving a quasiconvex function with a double-well structure in the geometrically linear elasticity setting. We show that under small dead-load perturbations, the Neumann boundary value problem has at least three solutions, a global minimizer, a local minimizer and a mountain pass solution. We show that our variational integral satisfies a Weak Palais-Smale condition (WPS) hence the mountain pass lemma applies.     

Gluing a mountain pass solution to a minimum

This paper presents a variational method for constructing solutions of a pendulum model equation that shadow a mountain pass solution glued to a minimum of the associated functional. It allows for more degenerate situations and gives more qualitative information than the classical Poincare-Birkhoff-Smale theory. Dedicated to the memory of Jürgen Moser     

Travelling wave solutions for the discrete sine-Gordon equation with nonlinear pair interaction

The focus of study is the nonlinear discrete sine-Gordon equation, where the nonlinearity refers to a nonlinear interaction of neighbouring atoms. The existence of travelling heteroclinic, homoclinic and periodic waves is shown. The asymptotic states are chosen such that the action functional is finite. The proofs employ variational methods, in particular a suitable concentration-compactness lemma combined with direct minimisation and mountain pass arguments.     

Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term

The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.     

Higher Critical Points in an Elliptic Free Boundary Problem

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.     

Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs

Variational gluing arguments are employed to construct new families of solutions for a class of semilinear elliptic PDEs. The main tools are the use of invariant regions for an associated heat flow and variational arguments. The latter provide a characterization of critical values of an associated functional. Among the novelties of the paper are the construction of “hybrid” solutions by gluing minima and mountain pass solutions and an analysis of the asymptotics of the gluing process.