We prove some new Hardy type inequalities on the bounded domain with smooth boundary in the Carnot group. Several estimates of the first and second Dirichlet eigenvalues for the p-sub-Laplacian are established.

This paper studies a nonlinear diffusion system with coupled nonlinear boundary flux and two kinds of inner sources (positive for the first and negative for the second), where the four nonlinear mechanisms are described by eight nonlinear parameters. The critical exponent of the system is determined by a complete classification of the eight nonlinear parameters, which is represented via the characteristic algebraic system introduced to the problem.

We study exponential decay property of radial ground states to a class of N-Laplacian elliptic equations in the whole space R^N. Their decay rates as |x|→∞ are obtained explicitly.

In this paper, we establish the existence of traveling wave solutions to the nonlinear three-dimensional viscoelastic system exhibiting long range memory. Under certain hypotheses, if the speed of propagation is between the speeds determined by the equilibrium and instantaneous elastic tensors, then the system has nontrivial traveling wave solutions. Moreover, the system has only trivial traveling wave solution in some cases.

Motivated by the study on the spontaneous potential well-logging, this paper deals with the homogenization of boundary conditions for a class of elliptic problems with jump interface conditions.

In this paper we consider the Cauchy problem for a semi-linear system of wave equations with Hamilton structure. We prove the existence of global smooth solution of the systemfor subcritical case by using conservation of energy and Strichartz's estimate. On the basis ofMorawetz-Poho\check{z}ev identity, we obtain the same result for the critical case.