In this paper, we first investigate the classification of positively homogeneous equations $(\phi_p(u'))'+q(t)\phi_p(u)=0$, $u(0)=0=u(1)$, where $p>1$ is fixed, $\phi_p(u)=|u|^{p-2}u$ and $q\in L^{\infty}(0,1)$, and then discuss the existence of solutions for non-homogeneous equations. The main method of classification is by using a generalized Prufer equation

$$ \theta'=|\4\cos_p\theta|^p+\frac{q(t)}{p-1}|\4\sin_p\theta|^p\quad\text{for }t\in(0,1), $$

where $\sin_p:\mathbb{R}\to[-1,1]$ is a periodic function and $\cos_pt=\mathrm{d}\sin_pt/\mathrm{d} t$ for $t\in\mathbb{R}$.

Let $\tau_\varOmega$ denote the lifetime of Brownian motion in an open connected set $\varOmega\subset\mathbb{R}^m$. We obtain the asymptotic behaviour of the expected lifetime $\mathbb{E}_x^y[\tau_\varOmega]$ as $y\to x$, where the Brownian motion is conditioned to start at $x$ and to exit $\varOmega\setminus\{y\}$ at $\{y\}$.

The main purpose of this paper is to study the mean value properties of certain Hardy sums over a short interval by using the mean-value theorems of the Dirichlet $L$-functions. Our main result is a mean-value formula for these sums.

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 are interested in the steady-state Euler–Poisson system for a potential flow used in the mathematical modelling of plasmas and semiconductors. In the case of the quasi-neutral limit, boundary layers can appear. We study this limit by using an asymptotic expansion. We show the existence and uniqueness of each profile and give the justification of the asymptotic expansion up to any order.

It is proved that a periodically forced second-order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a $T$-periodic stable solution for high values of the mean value of the forcing term. The method of proof combines a rescaling argument with the analysis of the first twist coefficient of the Birkhoff normal form for the Poincaré map.

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.

This paper is concerned with solutions of the complex Korteweg–de Vries (KdV) equation. We achieve two goals. First, we prove local well-posedness results for the complex KdV equation on a line, in a periodic domain and in a finite domain. These results are in line with the local well-posedness theory for the real KdV equation. Second, we establish a rigorous connection between the local (in space) regularity of the real part and that of the imaginary part of any solution to the complex KdV equation. This result partly validates the numerical observation that the real and imaginary parts of a singular solution of the complex KdV equation blow up at the same point and at the same time.