含临界指数与Hardy项耦合椭圆系统的非平凡解

研究了一类含临界指数耦合非线性项的奇异椭圆方程组,通过对临界耦合非线性项的分析与精确的能量估计,利用环绕定理,得到了这类方程组非平凡解的存在性.     

Multibump solutions for discrete periodic nonlinear Schrödinger equations

In this paper, we study the existence of multibump solutions for discrete nonlinear Schrödinger equations with periodic potentials. We first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group. Then, we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear Schrödinger equations, and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case. As an application, we get, without any symmetry assumptions, infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity.     

Solvability of nonlinear elliptic boundary value problem via its associated linear problem

We investigate the existence of solutions of a nonlinear elliptic boundary value problem at resonance. Under the condition that the associated linear boundary value problem has no sign-changing solution or nontrivial solution and some other additional conditions, we prove the existence of solutions or nontrivial (even multiple) solutions to the nonlinear problem. Thus the existence of solutions can be obtained when the nonlinearity may cross any finite number of eigenvalues of the linear problem.     

Multiplicity of positive solutions to weighted nonlinear elliptic system involving critical exponents

This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of R.     

一类非线性常微分方程的非平凡解

本文论讨了三个问题：①二阶非线性微分方程的初值问题；②运用非线性算子证明了非平凡解的存在性和唯一性；③给出了一个非负非平凡解的估计式．     

Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.     

Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

The main result of the paper concerns the existence of nontrivial exponentially decaying solutions to periodic stationary discrete nonlinear Schrödinger equations with saturable nonlinearities, provided that zero belongs to a spectral gap of the linear part. The proof is based on the critical point theory in combination with periodic approximations of solutions. As a preliminary step, we prove also the existence of nontrivial periodic solutions with arbitrarily large periods.     

含临界指数p-Kirchhoff型方程的非平凡解

本文研究了一类含临界指数的p-Kirchhoff型方程.利用变分方法与集中紧性原理,通过证明对应的能量泛函满足局部的(PS)_c条件,得到了这类方程非平凡解的存在性,推广了关于Kirchhoff型方程的相关结果.     

一个带Hardy项的椭圆方程的多重解

通过对偶变分方法证明了一个带Hardy项和临界非线性的非线性椭圆方程的非平凡解的存在性和多重性.     

Liouville-Type Theorem for a Nonlinear Degenerate Parabolic System of Inequalities

In this paper, we establish some new Liouville-type results for solutions of nonlinear degenerate parabolic system of inequalities. Nonexistence of nontrivial global solutions to initial-value problems is studied by using scaling transformations and test functions.     

Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

In this paper we study boundary value problems for semilinear equations involving strongly degenerate elliptic differential operators. Via a Pohozaev??s type identity we show that if the nonlinear term grows faster than some power function then the boundary value problem has no nontrivial solution. Otherwise when the nonlinear term grows slower than the same power function, by establishing embedding theorems for weighted Sobolev spaces associated with the strongly degenerate elliptic equations, then applying the theory of critical values in Banach spaces, we prove that the problem has a nontrivial solution, or even infinite number of solutions provided that the nonlinear term is an odd function.     

Existence of secondary bifurcations or isolas for PDEs

In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas of steady state solutions for parameter dependent nonlinear partial differential equations. The technique combines the Global Bifurcation Theorem, knowledge about the non-existence of nontrivial steady state solutions at the zero parameter value and explicit information about the coexistence of multiple nontrivial steady states at a positive parameter value. We apply the method to the two-dimensional Swift-Hohenberg equation.     

计算一类反应扩散方程分歧问题的Fourier配点法

1引言 关于反应扩散方程的研究由来已久,特别是对一些含参数的非线性反应扩散方程,由于其多解性和丰富的分歧现象,经常受到人们的关注.本文考虑如下非线性反应扩散方程组 {ut=γf(u,v)+uxx, vt=γg(u,v)+dvxx, (1) 相应的边界条件为 ux(t,0):ux(t,π)=vx(t,0)=vx(t,π)=0. (2) 我们选取Gierer-Meinhardt模型[1,2]为研究对象,即 {f(u,V)=a-bu+u2/v, g(u,v)=u2-v, 其中a、b和γ是正常数,d为参数.     

Hammerstein非线性积分方程组的非平凡解及应用

该文利用拓扑方法和锥理论研究下列Hammerstein非线性积分方程组u(x)=∫_G k(x,y)f(y,u(y),v(y)) dy,v(x)=∫_Gk(x,y)g(y,u(y),v(y)) dy.在适当的条件下,证明了上述方程组非平解的存在性,并把所得结果应用于研究非线性二阶常微分方程组边值问题的非平凡解的存在性.     

Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.     

Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is (p?1)-superlinear but does not satisfy the Ambrosetti–Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann Inst H Poincaré Anal Non Linéaire 8(1):43–57, 1991). Subsequently, by imposing additional conditions on the nonlinearity \({f(x,\cdot)}\), we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of (p, 2)-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters \({\lambda > 0}\) sufficiently small where one solution vanishes in the Sobolev norm as \({\lambda \to 0^+}\) and the other one blows up (again in the Sobolev norm) as \({\lambda \to 0^+}\).     

High energy solutions and nontrivial solutions for fourth-order elliptic equations

In this paper, we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth-order elliptic equations in RN via variational methods. Three main theorems are obtained.     

Nontrivial solutions of a nonlinear heat flow problem via Sperner’s Lemma

In this paper we investigate the existence of multiple nontrivial solutions of a nonlinear heat flow problem with nonlocal boundary conditions. Our approach relies on the properties of a vector field on the phase plane and utilizes Sperner’s Lemma, combined with the continuum property of the solutions funnel.     

Some global results for nonlinear fourth order eigenvalue problems

In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.     

Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method

We use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.