Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty }$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function....

We consider the focusing nonlinear Schrödinger equations $i{\partial }_{t}u+\Delta u+u{\left|u\right|}^{p-1}=0$. We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.

On considère dans cet exposé le modèle asymptotique “shallow-water/shallow-water" obtenu dans [3] à partir du système d’Euler à deux couches avec fond plat et toit rigide pour décrire la propagation d’ondes internes de grande amplitude. En dimension d’espace un, ce système est de type hyperbolique et la théorie locale du problème de Cauchy ne pose pas de difficultés majeures, même si d’autres questions (explosion en temps fini, perte d’hyperbolicité) s’avèrent délicates. En dimension deux d’espace...

We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data $u_0^{ϵ}$ for which the jacobian $J{u}_0^{ϵ}$ concentrates around an (oriented) rectifiable $m-2$ dimensional set, say ${\Gamma }_0$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t\>0$ continues to concentrate around some codimension $2$ submanifold, say ${\Gamma }_t$, and that the family $\left\{{\Gamma }_t\right\}$ of submanifolds evolves by binormal mean curvature flow. We prove...

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where $N\ge 4;V,K,g$ are periodic in $x_j$ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K\>0$. If $0\<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant c, we show that this equation has a nontrivial solution.