On a class of elliptic problems with indefinite nonlinearities

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.     

Multiplicity of positive solutions to M-point boundary value problem of second order impulsive differential equations

In this paper, by using Avery-Peterson theorem on a convex cone, we consider the m-point boundary value problems for second order impulsive differential equations with the nonlinear term depending on the first order derivative, the multiplicity result of three positive solutions are obtained.     

Multiplicity Results for Second-order Generalized Sturm-Liouville Boundary Value Problems with Dependence on the First Order Derivative

By using fixed point theorems,we consider multiplicity of positive solutions for second-order generalized Sturm-Liouville boundary value problem,where the first order derivative is involved in the nonlinear term explicitly.We show the existence of multiple positive solutions for the problems.Example is given to illustrate the main results of the article.     

非线性项含一阶导数广义Sturm-Liouville边值问题的多个正解

By using fixed point theorems, we consider multiplicity of positive solutions for second-order generalized Sturm-Liouville boundary value problem, where the first order derivative is involved in the nonlinear term explicitly. We show the existence of multiple positive solutions for the problems. Example is given to illustrate the main results of the article.     

Multiplicity of sign-changing solutions for some four-point boundary value problem

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f$f$.     

Multiplicity results for some second-order four-point boundary-value problems

In this paper, by using a fixed point theorem of functional type in a cone, some new results for the multiplicity of positive solutions of second-order four-point boundary-value problems are obtained. The emphasis is put on the nonlinear term involved with the first-order derivative.     

奇异非线性边值问题的经典Agarwal-O'Regan方法

改进了奇异非线性边值问题的经典Agarwal-O'Regan方法.利用这个改进的方法建立了奇异非线性(p,n-p)共轭边值问题正解的局部存在性与多解性,其中允许非线性项关于时间和空间变元同时奇异.主要工具是锥拉伸与锥压缩型的Guo-Krasnosel'skii不动点定理和精确先验估计技巧.特别的,考察了非自治奇异非线性二阶、三阶、四阶共轭边值问题.     

A continuum of solutions to variational inequality with nonlinear constraints: Existence and simplicial algorithm

In this paper, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality with explicit nonlinear constraints. This result follows in a constructive way by designing a simplicial algorithm. The algorithm operates on a triangulation of the unbounded regions and generates a piecewise linear path of parametrized stationary points. Each point on the path is an approximate solution.     

On The Coupled Cahn-hilliard Equations

This paper is concerned with a system of nonlinear partial differential equations, in short, the coupled Cahn-Hilliard equations, which consists of a fourth order quasilinear parabolic equation and a second order quasilinear parabolic equation. This system was recently derived by Penrose and Fife and also by Alt and Pawlow to describe the non-isothermal phase separation of a two-component system. The global existence and uniqueness of classical solutions is proved. The results about the asymptotic behavior, as time goes to infinity, of solution and about the existence and multiplicity of solutions to the corresponding stationary problem, which is a nonlinear boundary value problem involving nonlocal term and constraints, are also obtained.     

Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities

In this paper we develop a critical point theory for nonsmooth locally Lipschitz functionals defined on a closed, convex set extending this way the work of Struwe (Variational Methods, Springer, Berlin, 1990). Through a deformation result, we obtain minimax principles producing critical points. Then we use the theory to obtain positive and negative solutions of nonlinear and semilinear hemivariational inequalities. In this context we improve a result on positive solutions for semilinear elliptic problems due to Nirenberg (Variational methods in nonlinear problems, in: Topics in Calculus of Variations, Lecture Notes in Mathematics, vol. 1365, Springer, Berlin, 1987).     

Existence and Multiplicity Results for the p-Laplacian with a p-Gradient Term

We study the existence and multiplicity of positive solutions to p-Laplace equations where the nonlinear term depends on a p-power of the gradient. For this purpose we combine Picone’s identity, blow-up arguments, the strong maximum principle and Liouville-type theorems to obtain a priori estimates. Sebastián Lorca: Supported by FONDECYT N ^o 1080500.     

Unbounded connected component of the positive solution set of nonlinear operator equation and its applications

In this paper, by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters. As a result, we obtain some existence results for positive solutions of the nonlinear operator equation. The main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone.     

Remarks on nonlocal boundary value problems at resonance

We show that it is important to allow the nonlinear term to change sign when discussing existence of a positive solution for multipoint, or more general nonlocal, boundary value problems in the resonant case. When the nonlinear term has a fixed sign we obtain simple necessary and sufficient conditions for the existence of positive solutions.     

Nontrivial solutions of nonlocal boundary value problems for nonlinear higher-order singular fractional differential equations

This paper deals with the existence and multiplicity of nontrivial solutions of nonlocal boundary value problems for nonlinear higher-order singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples explain that our results cannot be obtained by the method of cone theory.     

Quasilinear elliptic problems under strong resonance conditions

In this paper we establish the existence and multiplicity of solutions for a quasilinear elliptic problem under strong resonance conditions at infinity. In order to control the resonance we consider a new hypothesis on the nonlinear term. In all results we use Variational Methods, Critical Groups and the Morse Theory.     

Cerami条件下脉冲边值问题古典解的存在性

在非线性项不满足Ambrosetti-Rabinowitz条件时研究脉冲微分方程边值问题,在原来的变分结构下,利用Cerami条件下成立的临界点理论来研究脉冲微分方程边值问题古典解的存在性和多重性.     

Structure of positive solution sets of semi-positone singular boundary value problems

In this paper we first obtain some results on the structure of positive solution sets of some differential boundary value problems. Then by using these results we obtain some existence and multiplicity results for positive solutions of differential boundary value problems. The method used to show the main results is that of global bifurcation theories.     

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.     

Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach

In this paper, we propose a modified generalized transformation for constructing analytic solutions to nonlinear differential equations. This improved unified ansätze is utilized to acquire exact solutions that are general solutions of simpler equations that are either integrable or possess special solutions. The ansätze is constructed via the choice of an integrable differential operator or a basis set of functions. The technique is implemented to obtain several families of exact solutions for a class of nonlinear evolution equations with nonlinear term of any order. In particular, the Klein–Gordon, the Sine–Gordon and Landau–Ginburg–Higgs equations are chosen as examples to illustrate the method.     

Existence and multiplicity of solutions to even order ordinary differential equations via variational methods

In this paper, we study the existence and multiplicity of solutions for two-point boundary value problems for even order ordinary differential equation. By using the critical point theory and the property of projective operator, we establish some conditions on the nonlinearity which guarantee that the nonlinear Hammerstein integral equation with quasi-positive definite kernel has at least one solution, two nontrivial solutions and finite pairs of solutions, respectively. As an application, we get the corresponding results for two-point boundary value problems for even order ordinary differential equation. Moreover, an example is presented to illustrate our main results.