On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.     

Implicit runge-kutta methods for differential inclusions

This paper is concerned with the application of implicit Runge-Kutta methods suitable for stiff initial value problems to initial value problems for differential inclusions with upper semicontinuous right-hand sides satisfying a uniform one-sided Lipschitz condition and a growth condition. The problems could stem from differential equations with state discontinuous right-hand sides. It is shown that there exist methods with higher order of convergence on intervals where the solution is smooth enough. Globally we get at least the order one.     

Three positive solutions of nonhomogeneous semilinear elliptic equations

In this paper, we use new analyses to assert that there are three positive solutions of Eq. (1.1) in infinite cylinder domain with hole .     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Hamiltonian identities for elliptic partial differential equations

New identities for elliptic partial differential equations are obtained. Several applications are discussed. In particular, Young's law for the contact angles in triple junction formation is proven rigorously. Structure of level curves of saddle solutions to Allen-Cahn equation are also carefully analyzed.     

Three positive solutions for nonlinear elliptic equations in finite strip with hole

In this paper, we study the effect of the domain on the multiplicity of positive solutions for the nonlinear elliptic equation. Using the Ekeland variational principle and the center mass function, we prove that there are at least three positive solutions for the nonlinear elliptic equation in finite strip with holes.     

A note on fuzzy differential equations

We study the effect of the forcing term to the solution of a fuzzy differential equation.     

The effect of domain shape on the number of positive and nodal solutions for semilinear elliptic equations

In this paper, we study the effect of domain shape on the number of positive and nodal (sign-changing) solutions for a class of semilinear elliptic equations. We prove a semilinear elliptic equation in a domain Ω$\Omega$ that contains m$m$ disjoint large enough balls has m²$m^2$ 2-nodal solutions and m$m$ positive solutions.     

Philos-type oscillation theorems for second order damped elliptic equations

Using general means, we establish several new Philos-type oscillation theorems for the second order damped elliptic differential equation
Σi,j=1 N Di[aij（x）Djy]＋Σi=1 N bi（x）Diy＋c（x）f（y）=0
under quite general assumptions. The obtained results are extensions of the well-known oscillation results due to Kamenev, Philos, Yan for second order linear ordinary differential equations and improve recent results of Xu, Jia and Ma.     

Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation

     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2.