A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

Multiplicity results for periodic solutions with prescribed minimal periods to discrete Hamiltonian systems

By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to discrete Hamiltonian systems are obtained.     

Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities

In this paper we study the nonlinear Kirchhoff equations on the whole space. We show the existence, non-existence, and multiplicity of solutions to this problem with asymptotically linear nonlinearities. This result can be regarded as an extension of the result in Li et al. (2012).     

Multiple positive solutions for a semilinear equation with prescribed singularity

Variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.     

Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations. The conditions for the existence of subharmonic solutions are established. Our results rectify some known results in the literature and will allow us, in the future, to deal with the subharmonic solutions for more extensive impulsive problems.     

The existence of periodic solutions for three-order neutral differential equations

In this paper, by using Kranoselskii fixed point point theorem and Mawhin''s continuation theorem, we establish two existence theorem on the periodic solutions for a class of three-order neutral differential equations.     

The minimal period problem for the classical forced pendulum equation

Considered in this paper is the existence/nonexistence of periodic solutions with prescribed minimal periods to the classical forced pendulum equation,

     

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

Multiple existence of non-radial solutions with group invariance for sublinear elliptic equations

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere.     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Concentration of homoclinic solutions for some fourth-order equations with sublinear indefinite nonlinearities

In this paper, we mainly explore the phenomenon of concentration of homoclinic solutions for a class of nonperiodic fourth-order equations with sublinear indefinite nonlinearities. The proof is based on variational methods.     

On the existence of solutions for the Schrödinger-Poisson equations

In this paper, we are concerned with the system of Schrödinger-Poisson equations

(*)