Related papers: Existence results for Toda systems with sign-chang…

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a…

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $\rho_1\in(0,4\pi)\cup(4\pi,8\pi)$ and $\rho_2\notin 4\pi\mathbb{N}.$ The key step is to derive the degree formula of the…

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h…

On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\beta \in [0,1)$ and $w_0(x) = |\log |x||^\beta $ then $$ \sup_{\int_B…

In this paper, we mainly focus on the existence of the viscosity solutions of \begin{equation*} \left\{ \begin{aligned} &H_1(x,Du_1(x),u_1(x),u_2(x))=0,\\ &H_2(x,Du_2(x),u_2(x),u_1(x))=0. \end{aligned} \right. \end{equation*} The standard…

In this paper we prove that the full symmetric Toda system is integrable in the sense of the Lie-Bianchi criterion, i.e. that there exists a solvable Lie algebra of vector fields of dimension $N=\dim M$ on the phase space $M$ of this system…

This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a…

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{da^2}{h(a^2)}+a^2d\,\sigma$ for some function $h$ where $d\sigma$ is the…

We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems \begin{equation} \begin{cases} -\Delta u+\lambda_1u=\mu_1u(e^{u^2}-1)+\beta…

In a recent paper we have considered the long time asymptotics of the periodic Toda lattice under a short range perturbation and we have proved that the perturbed lattice asymptotically approaches a modulated lattice. In the present paper…

In this paper we investigate the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for nonlinear Schr\"odinger systems of the form \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + r_1\beta…

In this paper, we construct invariant measures and global-in-time solutions for a fractional Schr\" odinger equation with a Moser-Trudinger type nonlinearity $$ i\partial_t u= (-\Delta)^{\alpha}u+ 2\beta u e^{\beta…

We establish an improved version of the Moser-Trudinger inequality in the hyperbolic space $\mathbb H^n$, $n\geq 2$. Namely, we prove the following result: for any $0 \leq \lambda < \left(\frac{n-1}n\right)^n$, then we have $$…

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(\sigma_1,\sigma_2)$ at each blowup point has the expression $$\sigma_i=2(N_{i1}\mu_1+N_{i2}\mu_2+N_{i3}),$$…

The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as…

High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics…

We are concerned with sign-changing solutions of the following gauged nonlinear Schr\"{o}dinger equation in dimension two including the so-called Chern-Simons term \begin{align*} \left\{ \begin{array}{ll} -\triangle {u}+\omega…

In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems \begin{equation*} H=h(y)+f(x,y,t), \end{equation*} where $y\in D\subseteq\mathbb{R}^n$ with $D$ being a closed bounded domain,…

In this work,we show the long time existence of smooth solu- tions to semigeostrophic equations on a torus when the initial dual density is bounded between two positive constants and smooth.The key ingredient is a more precise estimate on C…

On a smooth bounded 2-dimensional domain $\Omega$ we study the heat flow $u_t=\Delta u +\lambda (t)ue^{u^2}$ ($\lambda(t)$ is such that $d/dt ||u(t,\cdot)||_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe to investigate the…