Nodal solutions for a fourth-order two-point boundary value problem

We consider boundary value problems of fourth-order differential equations of the form

     

Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains

In this paper, we consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, K(x) is a continuous function. When Ω and K(x) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0<λ<λ₁, where λ₁ is the first Dirichlet eigenvalue of on Ω.     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

Behavior at infinity of solutions of an equation of Osserman-Keller type

We study the behavior at infinity of solutions of equations of the form Δu=u^p, where p>1, in dimensions n?3. In particular we extend results proved by Loewner and Nirenberg in Contribution to Analysis, 1974, pp. 245-272 for the case p=(n+2)/(n−2), n?3, to values of p in the range p>n/(n−2), n?3.     

New travelling wave solutions for a nonlinearly dispersive wave equation of Camassa-Holm equation type

In this paper, the integral bifurcation method is used to study a nonlinearly dispersive wave equation of Camassa-Holm equation type. Loop soliton solution and periodic loop soliton solution, solitary wave solution and solitary cusp wave solution, smooth periodic wave solution and non-smooth periodic wave solution of this equation are obtained, their dynamic characters are discussed. Some solutions have an interesting phenomenon that one solution admits multi-waves when parameters vary.     

一类拟线性椭圆方程整体解的不存在性

这篇文章主要利用常微分技术讨论了一个二阶拟线性椭圆方程Lpu≡div（｜Du｜^p-2Du）=f（x,u）,x∈R^N的整体解的不存在性.我们只考虑2≤p〈N的情況并且在假设f（x,u）关于第二个变量u没有单调性的情况下得到整体解的不存在性结果.     

Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation

In this paper, using the mountain pass theorem, we give the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations.     

Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

In this paper we study existence and multiplicity of weak solutions of the homogenous Dirichlet problem for a singular semilinear elliptic equation with a quadratic gradient term. The proofs for the main results are based on a priori estimates of solutions of approximate problems.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Nodal solutions of boundary value problems of fourth-order ordinary differential equations

We study the existence of nodal solutions of the fourth-order two-point boundary value problem

     

On sign-changing solution for a fourth-order asymptotically linear elliptic problem

In this paper we deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.     

Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.     

Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity

In this paper, we study the existence of ground state solutions of nonlinear elliptic equation with logarithmic nonlinearity by the Linking theorem and logarithmic Sobolev inequality. Our results are quite different from those in the case of polynomial nonlinearity.     

带有类p-Laplacian算子的Kirchhoff型方程的多重解

研究一类带有类p-Laplacian算子的Kirchhoff型方程.利用对称山路定理,得到了多重解存在的充分条件,推广和改进了已有结果.     

A note on rational solutions to a Hirota-Satsuma-like equation

With the generalized bilinear operators based on a prime number $p=3$, a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.     

A fourth-order boundary value problem for a Sturm-Liouville type equation

An existence result of multiple solutions for a fourth-order Sturm-Liouville boundary value problem with variable parameters is established. As a consequence, three solutions for a boundary value problem with a fourth-order equation in a complete form are obtained. Our approach is based on variational methods.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.