We establish in this paper existence results for critical strongly indefinite semilinear elliptic systems defined on both bounded domains and $\mathbb R^N$.

The subject of this paper is the behavior of the director field of a nematic liquid crystal in flow through a tube with circular cross-section. Both the flow and the director field are assumed to have cylindrical symmetry. The requirement of finite Frank-Oseen energy forces ``admissible" director fields to be axially directed at the location of the symmetry axis. Thus, the angle between the axis and the director field at the location of the axis amounts to $k\pi$, $k$ being an integer. In the steady case, it is shown that $k$ is largely (but not uniquely) determined by the (Dirichlet) boundary conditions. In particular, this may give rise to line singularities with finite energy density. Moreover, the associated Dirichlet problem may have several distinct solutions with identical value of $k$. Analogously to the heat flow of harmonic mappings, finite-time blow-up phenomena for the associated parabolic problem are established.

This paper is devoted to the rigorous analysis of the dynamics of single-spike solutions in the shadow Gierer-Meinhardt system. We derive a system of ODEs which governs the motion of the spike. Careful analysis of this system shows that the spike moves exponentially slowly towards the point on the boundary that is the closest to the spike.

Nonlinear elliptic systems with $p$-structure for $p \! \geq \! 2$, such as \[ -\mbox{div}\left((1+|\nabla u|^{p-2})\nabla u\right) =f(x)+\sum_{i=1}^n\partial_if_i(x) ,\] are considered under mixed boundary value conditions on nonsmooth domains. Regularity results in fractional-order Sobolev spaces are proven, e.g., $u\in W^{r,p}(\Omega)$ for all $r <1+\frac{1}{p}$ and $|\nabla u|^p\in W^{s,1}(\Omega)$ for some $s>1$.