system (1.1) is completely integrable, which is well-known in integrable systems theory. The Liouville equation and the Toda system arise in many physical models. In Chern-Simons theories, the Liouville equation is closely related to Abelian models, while the Toda system is related to non-Abelian models. See for instance the books [10] and [24] and the references therein. Though the Liouville e...

The asymptotic behaviour of a class of inhomogeneous scalar field cosmologies with a Liouville type of potential is studied. We define a set of new variables for which the phase space of the system of Einstein equations is bounded. This allows us to perform a complete analysis of the evolution of these cosmologies. We also discuss the extension of the cosmic no-hair theorem.

Motivated by Carleman’s proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.

We determine eigenvalue intervals of λ1 and λ2 for the existence of at least one positive solution for a coupled system of Riemann–Liouville type multi-point fractional order boundary value problems by utilizing a fixed point theorem on a cone under suitable conditions.

In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescali...

Abstract Inequalities play a major role in pure and applied mathematics. In particular, the inequality plays an important study of Rosseland’s integral for stellar absorption. this paper we obtain new Milne-type inequalities, apply them to generalized Riemann–Liouville-type operators, which include most known Riemann–Liouville operators.

It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive water wave system, and auto-Bäcklund transformations for coupled three-component KdVtype ...

In this paper the existence of unique positive solutions for system (p,q,r)-Lapalacian Sturm-Liouville type two-point fractional order boundary vaue problems is established by an application n-fixed point theorem ternary operators on partially ordered metric spaces.