Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems

In the present paper,the author studies the existence of sign-changing solutions for nonlinear elliptic equations,which have jumping nonlinearities,and may or may not be resonant with respect to Fu(?)ik spectrum,via linking methods under Cerami condition.     

跳跃非线性条件下椭圆边值问题的变号解和多解存在性

本文应用Banach空间常微分方程和极大极小理论,特别是相对山路引理研究了零点和无穷远点跳跃非线性条件下椭圆边值问题的变号解和多解,得到新的变号解和多解存在性定理,最后我们得到6个非平凡解的存在性.     

Sign-Changing and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions. Received April 21, 1998, Revised November 2, 1998, Accepted January 14, 1999     

Existence of nontrivial solutions for Kirchhoff-type problems with jumping nonlinearities

In this study, the Fučik spectrum is used to investigate Kirchhoff-type problems with jumping nonlinearities at infinity, and a new result on the existence of nontrivial solutions is obtained.     

Fucik spectrum, sign-changing and multiple solutions for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity

In this paper, Fucik spectrum, ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and some new results on the existence of nontrivial solutions, multiple solutions and sign-changing solutions are obtained. In one case seven nontrivial solutions are got. The techniques have independent interest.     

非线性算子方程变号解的存在性及其应用

本文利用锥理论讨论了非线性算子方程变号解的存在性,并将抽象结果应用于Sturm-Liouville两点边值问题。所得结果无论在理论上还是在应用上都是新的.     

Fucik Spectrum, Sign-Changing, and Multiple Solutions for Semilinear Elliptic Boundary Value Problems with Resonance at Infinity

In this paper, we use Fucik spectrum, ordinary differential equation theory of Banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, especially with resonance at infinity, and obtain new results on the existence of multiple solutions and sign-changing solutions with a weakening of the P.S. condition. In one case we get up to seven nontrivial solutions. The techniques have independent interest.     

Existence and Multiplicity of Periodic Solutions of the Second-Order Differential Equations with Jumping Nonlinearities

We provide sufficient conditions for the existence and multiplicity of periodic solutions for Duffing's equations with jumping nonlinearities under resonance conditions. Received March 1, 1999, Revised September 16, 1999, Accepted February 1, 2000     

On the Generalization of the Courant Nodal Domain Theorem

In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if u_λn is an eigenfunction associated with the nth variational eigenvalue, λ_n, then u_λn has at most 2n−2 nodal domains. Also, if u_λn has n+k nodal domains, then there is another eigenfunction with at most n−k nodal domains.     

A global characterization of the Fu?ík spectrum for a system of ordinary differential equations

The Fu?ík spectrum for systems of second order ordinary differential equations with Dirichlet or Neumann boundary values is considered: it is proved that the Fu?ík spectrum consists of global C¹ surfaces, and that through each eigenvalue of the linear system pass exactly two of these surfaces. Further qualitative, asymptotic and symmetry properties of these spectral surfaces are given. Finally, related problems with nonlinearities which cross asymptotically some eigenvalues, as well as linear-superlinear systems are studied.     

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

Using a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u^″+bu⁺−au⁻=p(t). The choice of the equation is motivated by the studies about the Dancer-Fu?ik spectrum and the Lazer-McKenna suspension bridge model.