We are concerned with stability problems for a holonomic system $\mathcal {S}$ with $n$ degrees of freedom having $n-m$ cyclic coordinates, $m<n$. Let $x$ be the set of the acyclic coordinates and let $v$ be the set of the generalized velocities corresponding to the cyclic coordinates. For $\mathcal {S}$ conservative or subject to a dissipation restricted to the acyclic coordinates, we revisit classical stability results concerning the steady motions of the system and give some new contribution. When $\mathcal {S}$ is strictly dissipative with respect to all the coordinates, the integrals of momenta disappear and so do the steady motions. In this case, under suitable conditions there exist motions for which $x$ is constant and, consequently, $v \rightarrow 0$ as $t \rightarrow +\infty$ (pseudosteady motions). We analyze the stability properties with respect to $(x, \dot x)$ of these motions. Such properties define a stable or unstable behavior with respect to $(x, \dot x)$ of corresponding steady motions of the conservative system under the influence of strictly dissipative perturbing

In this paper we provide an existence result for an implicitly given elliptic differential equation of the form $f(x,u,Lu)=0$ under Dirichlet boundary conditions. The peculiarity of this implicit equation is that the function $f$ may be discontinuous in all its arguments. The main tool used to treat this problem is a fixed point result in partially ordered sets based on a generalized iteration method combined with an appropriately modified method of upper and lower solutions.

Symmetry properties of positive solutions of the equations $$ \Delta u + \phi(|x|)f(u) = 0 $$ in $\mathbb{R}^n$ are considered. We employ the moving plane method based on the maximum principle on unbounded domains to obtain new results on symmetr

This paper studies the nonnegative weak solutions of a porous medium equation with strong absorption. We prove an apriori $\text{L}^{\infty}$ estimate through Moser iteration and obtain a compactness theorem and an integral-type Harnack inequality. Using these fundamental results we prove the existence of initial traces of weak solutions and obtain the existence of a fundamental solution and the nonexistence of a very singular solution, as byproducts. As an another application of our apriori estimates we prove the finiteness of the propagation speed without using comparison principle

We study the finite-energy solutions of a generalization of the classical Ginzburg-Landau system (problem (0.1)). In the first part of this paper we consider the solutions of (0.1) satisfying $K=M=2$ and $\int_{\mathbb{R}^2} P_n(\vert u \vert^2) <\infty$. We establish a phenomenon of quantization for the "mass" $\int_{\mathbb{R}^2} P_n(\vert u \vert^2)$ that genera\-lizes a well-known result of Brezis, Merle and Rivière for the classic Ginzburg-Landau system of equations in $\mathbb{R}^2$. The second part is devoted to solutions of (0.1) satisfying $\int_{\mathbb{R}^K} \vert\nabla u\vert^2 < \infty$. We establish some Liouville-type results and in particular we prove that any locally $L^3$ solution of the Ginzburg-Landau system $-\Delta u = u(1-\vert u \vert^2)$ in $\mathbb{R}^K$ (K$>$1) satisfying $\int_{\mathbb{R}^K}\vert \nabla u\vert^2 <\infty$, is a constant function. We also obtain that any solution of (0.1) with finite energy (0.4) is a constant. In the last section we prove that any solution $u$ of $(0.1)$ is smooth and satisfies $\vert u \vert^2 \le k_n $ on $\mathbb{R}^K$.

The global well-posedness of the initial-value problem associated to the Korteweg-de Vries equation with bore-like data is studied. In particular, we show that in the Sobolev space $H^s$, $s\ge 2$, the solutions of this problem remain bounded for any time. We also establish similar results for solutions of the initial-value problem associated to the Benjamin-Ono equation with this kind of data.