In this paper we study the following nonlinear elliptic problem with Dirichlet boundary condition: $-\Delta u =K(x)u^p$, $u>0$ in $\Omega$, $u =0$ on $ \partial \Omega$, where $\Omega$ is a bounded, smooth domain of $\mathbb R^n$, $n\geq 4$ and $p+1=2n/(n-2)$ is the critical Sobolev exponent. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational problem, we prove some existence results.

We introduce a new notion of renormalized dissipative solutions for a scalar conservation law $u_{t}+\mathrm{div}\, {\mathrm{\mathbf{F}}}(u)=f$ with locally Lipschitz ${\mathrm{\mathbf{F}}}$ and $L^{1}$ data, and prove the equivalence of such solutions and renormalized entropy solutions in the sense of Benilan et al. The structure of renormalized dissipative solutions is more useful in dealing with relaxation systems than the renormalized entropy scheme. As an example, we apply our result to contractive relaxation systems in merely an $L^{1}$ setting and construct a renormalized dissipative solution via relaxation.

This paper addresses the basis property of a linear hyperbolic system with dynamic boundary condition in one space variable whose general form was first studied in [15]. It is shown that under a regularity assumption, the spectrum of the system displays a distribution on the complex plane similar to zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis for its root subspace. The state space thereby decomposes into a topological direct sum of the root subspace with another invariant subspace in which the associated semigroup is supperstable: that is to say, the semigroup is identical to zero after a finite time. As a consequence, the spectrum-determined growth condition is established.

A wave equation in the curl-free space is treated, and its weak solutions are constructed by use of a minimizing method. Numerical calculations are also done. Physical backgrounds are on equilibrium states of the smectics liquid crystal. Motion of zero points for the director vector is shown under control of an equation of wave type.

In this paper, we consider Chua's circuit: $$ \varepsilon u'=z+f(u) , \ \ z'=u+w-z, \ \ w'=-\beta z-\gamma w, $$ where $f(u)$ is chosen as a cubic function, $\beta>0$ and $\gamma\geqslant 0$ are constants, and $\varepsilon>0$ is a small parameter. We prove that the flow defines a Poincar$\acute{e}$ map from a compact set which is homeomorphic to the unit disk to itself and then apply Brouwer's fixed-point theorem to conclude that the system has a ``big'' periodic solution. This global analysis is viewed as a step towards understanding chaos in this model analytically.