We study Halphen's equation and provide solutions in terms of elliptic functions of the second kind. The connection between Halphen's equation and algebro-geometric solutions of the Boussinesq hierarchy is discussed.

We deal with an obstacle problem with measure data. We improve our previous result of existence in a form which permits us to obtain a convergence result with respect to Mosco convergence of obstacles and uniqueness of the solution in the case where the measure data vanishes on sets of zero capacity. We also give a property, which is analogous, for renormalized solutions, to a recent characterization obtained in [10].

We obtain existence of a weak solution of the equations of zero-pressure gas dynamics, in any number of space dimensions plus time, given initial data of finite mass and kinetic energy. Our solution satisfies the "sticky particle" or adhesion dynamics conditions. The argument depends on a definition of a weak solution based on the introduction of Lagrangian coordinates and in particular on an application of the entropy condition proposed by C. Dafermos for hyperbolic systems of conservation laws.

We establish the existence of nodal solutions for the problem \begin{eqnarray*} -{\varepsilon}^2 \Delta u + u = |u|^{p-2} u \quad \hbox{in}\, \Omega ,\ \ \ u \in H_0^1 (\Omega ), \end{eqnarray*} where $\Omega $ is a bounded domain, $2 <p <2N/(N-2)$ for $N\geq 3,$ and $\ 2 <p <\infty$ for $N=2$. It is shown that, corresponding to each pair of points $P_1, P_2 \in (\Omega \times \Omega )$, whose location depends on the geometry of $\Omega $, there is a nodal solution, which has, for small ${\varepsilon}$, exactly one positive and one negative peak, and that the peak points converge, as ${\varepsilon} \to 0$, to $(P_1, P_2)$.

We give a variational characterization of the regions of stability and instability for planar Hamiltonian systems which are periodic in the time dependence, and convex in the momentum variables. In particular, we give necessary and sufficient conditions for stability, in terms of the Morse index of the variational problem with periodic or anti-periodic boundary conditions. We also show that this index coincides with the Maslov index of a curve of one-dimensional lines in the space $\mathbb{R}^2$. This result generalizes a result of Poincaré (1892) on unstable closed geodesics on orientable surfaces which minimize the arc length functional, and also gives a new test for stability based on the variational description of parametric resonance equivalent to that given by Liapunov (1892).

We consider semilinear boundary value problems of the form \begin{equation} L u(x) = f(x,u(x)) + h(x), \quad x \in (0,\pi), \tag*{(1)} \end{equation} where $L$ is a $2m$th-order, self-adjoint, disconjugate ordinary differential operator on $[0,\pi]$, together with appropriate boundary conditions at $0$ and $\pi$, while $f : [0,\pi] {\times} \mathbb R {\rightarrow} \mathbb R$ is a Carath{\'e}odory function and $h \in L^2(0,\pi)$. We assume that the limits $$ a(x) := \lim_{\xi {\rightarrow} \infty} f(x,\xi)/\xi, \quad b(x) := \lim_{\xi {\rightarrow} -\infty} f(x,\xi)/\xi, $$ exist for almost every $x \in [0,\pi]$ and $a,\,b \in L^{\infty}(0,\pi)$, but $a \ne b$. In this case the nonlinearity $f$ is termed {\em jumping}. Closely related to (1) is the "limiting" boundary value problem \begin{equation} L u = a u^+ - b u^- +{\lambda} u + h, \tag*{(2)} \end{equation} where $u^{\pm}(x) = \max\{\pm u(x),0\}$ for $x \in [0,\pi]$, and ${\lambda}$ is a real parameter. Values of ${\lambda}$ for which (2) (with $h=0$) has a nontrivial solution $u$ will be called {\em half-eigenvalues} of $(L;a,b)$. In this paper we show that a sequence of half-eigenvalues exists, with certain properties, and we prove various results regarding the existence and multiplicity of solutions of both (1) and (2). These result depend strongly on the location of the half-eigenvalues relative to the point ${\lambda}=0$. Some geometric properties of the Fučík spectrum of $L$ are also briefly discussed.