相关论文: The cone Moser-Trudinger inequalities and their ap…

This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands…

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R…

We consider weak solutions of the equation $$-\Delta_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } \Omega,$$ where $H$ is in some cases called Finsler norm, $\Omega$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and…

Sharp Moser-Trudinger type inequalities and their extremal functions play an important role in studying nonlinear PDEs and geometry. We establish a new sharp Moser-Trudinger type inequality in the upper half space in two dimensions and…

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure \[ \partial_t u - \div\big(M\nabla u+ E h(u)\big)=0, \] with $M$ a bounded elliptic matrix, $E$ a vector field in a suitable Lebesgue…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb R^{1+d}$. In the range for which…

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider…

We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces $\dot{W}^{1,p}_{0, \text{rad}}(B_R^N)\,(p<N)$. Our results give a revision of an…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…

We consider semilinear elliptic problems of the form \[ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), \] where $A\subset\mathbb{R}^N$, $N\geq3$, is either a bounded or unbounded annulus, and $\lambda \geq0$. We study a broad class of…

This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…

In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform…

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincar\'e disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned}…

This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schr\"odinger operators on bounded domains.

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the…

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,\alpha}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb…