Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

On a class of differential-difference equations arising in number theory

Supported in part by an NSF grant.     

On a class of non-uniformly elliptic equations

In this paper we study the existence and uniqueness of both weak solutions and entropy solutions for the Dirichlet boundary value problem of a class of non-uniformly elliptic equations. A comparison result is also discussed. Some well-known elliptic equations are the special cases of this equation.     

Multiple positive solutions for a class of nonlinear elliptic equations

Via delicate estimates, we characterize an exact growth order near zero for positive solutions of a class of nonlinear elliptic equations. Using this characterization, we obtain multiple positive solutions for equations involving critical nonlinearity.     

On a class of nonlinear elliptic equations with fast increasing weight and critical growth

We are concerned with the existence of rapidly decaying solutions for the equation

     

Capacitary estimates of solutions of a class of nonlinear elliptic equations

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$ and K a compact subset of $\text{?Ω}$. Assume that q?(N+1)/(N?1) and denote by U_K the maximal solution of ?Δu+u^q=0 in $\text{Ω}$ which vanishes on $\text{?Ω?K}$. We obtain sharp upper and lower estimates for U_K in terms of the Bessel capacity C_2/q,q′ and prove that U_K is σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\text{?Ω}$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Pointwise estimates in a class of fourth order nonlinear elliptic equations

Summary The subharmonicity of a function, which is defined on solutions of a fourth order nonlinear elliptic differential equation, leads to estimates on the solution and the gradient and Laplacian of the solution at interior points in terms of values on the boundary.

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Zusammenfassung Die subharmonische Eigenschaft von Funktionen, die durch die Lösungen von nichtlinearen elliptischen Differentialgleichungen vierter Ordnung bestimmt sind, führt zu punktweiser Abschätzung der Lösung, deren Gradienten, und Laplace Operatoren, abhängig von den Randwerten sind.

     

Some further results for a class of weighted nonlinear elliptic equations

Following our previous work [M. Lin, Multiple positive solutions for a class of weighted nonlinear elliptic equations, preprint, 2006], we give an exact growth order near zero for positive solutions of a class of weighted elliptic equations and use it to give the existence of multiple positive solutions and sign changing solutions of the considered problem.     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].     

Concentration and energy invariance for a class of fully nonlinear elliptic equations

We study the asymptotic behavior of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.