In this paper we study some very weak notions of differentiability arising in connection with the spatial regularity of flows associated with non-smooth vector fields. The main difference from other similar concepts, also studied in a Sobolev setting, is that the convergence of difference quotients has to be understood as convergence in measure. We show in particular that the classical approximate differentiability is a property strictly stronger than approximate differentiability in measure.

It is shown that, if $f$ is a meromorphic function of order zero and $q\in\mathbb{C}$, then

\begin{equation} \label{abstid} m\bigg(r,\frac{f(qz)}{f(z)}\bigg)=o(T(r,f)) \tag{\ddag} \end{equation}

for all $r$ on a set of logarithmic density $1$. The remainder of the paper consists of applications of identity \eqref{abstid} to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of $q$-difference equations. The results obtained include $q$-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

We study the behaviour at infinity, with respect to the spatial variable, of solutions to the magnetohydrodynamics equations in $\mathbb{R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free non-stationary Navier–Stokes equations when $|x|\to\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are new boundedness criteria for convolution operators in weighted spaces.

We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.

To prove that certain standard 2-complexes are aspherical we explore a strategy that combines a well-known method based on a graph theoretic lemma of Stallings with a process of reversing the orientation of edges in spherical diagrams. We apply this strategy to labelled-oriented-tree complexes and, more generally, to labelled-oriented-graph (LOG) complexes and obtain classes of aspherical LOG complexes to which, traditionally (without the reversing of edges), Stallings's lemma could not be applied to prove asphericity. These classes contain examples whose asphericity, as far as we know, could not be established by any previous method.

We investigate the asymptotic behaviour as $k\to+\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\varOmega)$ of solutions of the equations $\Delta^2u_k=V_k\mathrm{e}^{4u_k}$ on $\varOmega$, where $\varOmega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to+\infty}V_k=1$ in $C^0_{\mathrm{loc}}(\varOmega)$. The corresponding two-dimensional problem was studied by Brézis and Merle and Li and Shafrir, who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Robert and Struwe in 2006, such a quantization does not hold in dimension $4$ for the problem in its full generality. We prove here that, under a natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension $2$.

The main goal of this work is to study the existence and uniqueness of a positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.

This paper studies blowing-up properties of a unique positive principal eigenvalue for a linear elliptic eigenvalue problem with an indefinite weight function and Neumann boundary condition. Necessary and sufficient conditions for the blowing-up property are discussed, based on the variational characterization of the unique positive principal eigenvalue. A counterexample is constructed, which shows that a known necessary and sufficient condition for the blowing-up property in the Dirichlet boundary condition case no longer remains true in the Neumann case.

In this paper, we study the self-similar solutions and the time-asymptotic behaviour of solutions for a class of degenerate and singular diffusion equations in the form

$$ u_t=(|(p(u))_x|^{\lambda-2}(p(u))_x)_x,\quad -\infty<x<+\infty,\quad t>0, $$

where $\lambda>2$ is a constant. The existence, uniqueness and regularity for the self-similar solutions are obtained. In particular, the behaviour at two end points is discussed. Based on the monotonicity property of the self-similar solutions and the comparison principle, we also investigate the time convergence of the solution for the Cauchy problem to the corresponding self-similar solution when the initial data have some decay in space variable.

In this paper, we consider the following non-homogeneous elliptic problem

\begin{gather*} -\Delta u+u=\lambda(u^{p-1}+h(x))\text{ in }\varOmega, \\ u>0\text{ in }\varOmega,\quad u\in H_0^{1}(\varOmega), \end{gather*}

where $2<p<2N/(N-2)$, $N\geq3$, $\lambda>0$, $\varOmega=\bar\omega^{c}$ is an exterior domain in $\mathbb{R}^N$, where $\omega$ is a bounded set with smooth boundary, $h\in L^{2}(\varOmega)\cap L^{\beta}(\varOmega)$ ($\beta>\tfrac{1}{2}N$ if $N\geq4$ and $\beta=\tfrac{1}{2}N$ if $N=3$) is non-negative and $h(x)\not\equiv0$. We use variational methods to show that there exists a positive number $\lambda_0$ such that the equation above has at least two positive solutions if $\lambda\in(0,\lambda_0)$, no positive solution if $\lambda>\lambda_0$ and at least one positive solution if $\lambda=\lambda_0$. Furthermore, we use the Lyusternik–Schnirelman category to show that there exists a positive number $\lambda_{\ast}<\lambda_0$ such that the above equation has at least three positive solutions if $\lambda\in(0,\lambda_{\ast})$.

In this paper, we consider the existence and multiplicity of periodic solutions for the Duffing equation $x''+g(x)=p(t)$ with a singularity. When the time map has oscillating properties, $g(x)$ possesses a singularity at the origin and tends to $+\infty$ as $x\to+\infty$ and other conditions hold. We obtain the existence of harmonic solutions and the multiplicity of subharmonic solutions of the given equation by using the phase-plane analysis methods and the generalized Poincaré–Birkhoff twist theorem.

We consider the problem $-\text{div}(p(x)\nabla u)=\lambda{u}+\alpha|u|^{r-1}u$ in $\varOmega$, $\partial u/\partial\nu=Q(x)|u|^{q-2}u$ on $\partial\varOmega$, where $\varOmega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $q=2(N-1)/(N-2)$ and $2<r<q$. Under some conditions on $\partial\varOmega$, $p$, $Q$, $\lambda$, $\alpha$ and the mean curvature at some point $x_0$, we prove the existence of solutions of the above problem. We use variational arguments, namely the concentration–compactness principle, min–max principle and the mountain-pass theorem.