By a combination of variational and topological techniques in the presence invariant cones, we detect new type positive axially symmetric solutions Dirichlet problem for elliptic equation $$ -\Delta u + = a(x)|u|^{p-2}u an annulus $A \subset \mathbb{R}^N$ ($N\ge3$). Here $p>2$ is allowed to be supercritical $a(x)$ but possibly nonradial function with additional symmetry monotonicity properties,...

The security of several elliptic curve cryptosystems is based on the difficulty to compute the discrete logarithm problem. The motivation of using elliptic curves in cryptography is that there is no known sub-exponential algorithm which solves the Elliptic Curve Discrete Logarithm Problem (ECDLP) in general. However, it has been shown that some special curves do not possess a difficult ECDLP. I...

In this paper Lie’s formalism is applied to deduce classes of solutions of a nonlinear partial differential equation (nPDE) of second order with quadratic nonlinearity. The equation has the meaning of a field equation appearing in the formulation of kinetic models. Similarity solutions and transformations are given in a most general form derived to the first time in terms of reciprocal Jacobian...

We demonstrate the existence of elliptic vortices of electromagnetic scalar wave fields. The corresponding intensity profiles are formed by propagation-invariant confocal elliptic rings. We have found that copropagation of this kind of vortex occurs without interaction. The results presented here also apply for physical systems described by the (2+1) -dimensional Schrödinger equation.

By using the weighted averaging techniques, we establish oscillation criteria for the second order damped quasilinear elliptic differential equation

We prove a global Lorentz estimate of the Hessian of strong solutions to a class of asymptotically regular fully nonlinear elliptic equations over a C1,1 smooth bounded domain. Here, the approach of the main proof is based on the Possion’s transform from an asymptotically regular elliptic equation to the regular one.

In this paper we study fully nonlinear obstacle type problems in Hilbert spaces. We introduce the notion of Q-elliptic equation and prove existence, uniqueness, and regularity of viscosity solutions of Q-elliptic obstacle problems. In particular we show that solutions of concave problems with semiconvex obstacles are in the space W 2,∞ Q .

An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. More precisely, we show that a pair of elliptic functions (p, q) is an algebro-geometric AKNS potential, that is, a solution of some equation of the stationary AKNS hierarchy, if and only if the associated linear differential system JΨ +QΨ = EΨ, where J = (

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbit...

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1 +X)(1 + Y )(X + Y )− αXY , α ∈ R.