We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.

The paper is devoted to the estimate u(x,k)Kk{capp,w(F)pw(B(x,))} 1p-1, $2p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain $B(x_0,1)\setminus F$, $F\subset B(x_0,d)=\{x\in {ℝ}^n|x_0-x|<d\}$, $d<\frac{1}{2}$, under the boundary-value conditions $u(x,k)=k$ for $x\in \partial F$, $u(x,k)=0$ for $x\in \partial B(x_0,1)$ and where $0<\rho \le dist(x,F)$, $w\left(x\right)$ is a weighted function from some Muckenhoupt class, and $ca{p}_{p,w}\left(F\right)$, $w\left(B\right(x,\rho \left)\right)$ are weighted capacity and measure of the corresponding sets.

Cet exposé présente les résultats de l’article [3] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.

In this paper we consider two-dimensional quasilinear equations of the form $\text{div}\left(a\left(\left|\nabla u\right|\right)\nabla u\right)+f\left(u\right)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u$(notice that $\text{arg}\left(\nabla u\right)$ is a well-defined real function since $\left|\nabla u\right|>0$ on ${\mathbb{R}}^2$) we prove that $u$ is one-dimensional, i.e., $u=u\left(\nu \cdot x\right)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides a proof of...

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation$$\Delta u+\rho \left(\frac{e^u}{{\int }_T{e}^u}-\frac{1}{\left|T\right|}\right)=0,{\int }_{T}u=0.$$It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8\pi$. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting ${\lambda }_1\left(T\right)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le min\{8\pi ,{\lambda }_1\left(T\right)|T\left|\right\}$. Our results hold more generally for solutions that are Steiner symmetric, up to a translation....