Trudinger Moser Inequality Research Articles

Continuity of delta invariants and twisted Kähler–Einstein metrics

We show that delta invariant is a continuous function on the big cone. We will also introduce an analytic delta invariant in terms of the optimal exponent in the Moser–Trudinger inequality and prove that it varies continuously in the Kähler cone, from which we will deduce the continuity of the greatest Ricci lower bound. Then building on the work Berman–Boucksom–Jonsson, we obtain a uniform Yau–Tian–Donaldson theorem for twisted Kähler–Einstein metrics in transcendental cohomology classes.

A singular Moser-Trudinger inequality for mean value zero functions in dimension two

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partial\Omega$. In this paper, we prove that for any $\beta\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(\Omega), \int_\Omega u dx=0, \int_\Omega|\nabla u|^2dx\leq1}\int_\Omega \frac{e^{2\pi(1-\beta) u^2}}{|x|^{2\beta}}dx$$ is finite and can be attained. This partially generalizes a well-known work of Alice Chang and Paul Yang (J. Differential Geom. 27 (1988), no. 2, 259-296) who have obtained the inequality when $\beta=0$.

Singular Moser–Trudinger Inequality Involving $$L^{n}$$ Norm in the Entire Euclidean Space

Singular Moser–Trudinger Inequality Involving $$L^{n}$$ Norm in the Entire Euclidean Space

Finite entropy vs finite energy

Probability measures with either finite Monge-Ampere energy or finite entropy have played a central role in recent developments in Kahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Ampere measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.

Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group

Sharpened Trudinger–Moser Inequalities on the Euclidean Space and Heisenberg Group

Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface

Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang [7] and the first named author [32], and complements Fontana's inequality of two dimensions [15]. The blow-up analysis in the current paper is far more elaborate than that of [32], and particularly clarifies several ambiguous points there. In precise, we prove the existence of isothermal coordinate systems near the boundary, the existence and uniform estimates of the Green function with the Neumann boundary condition. Also our analysis can be applied to the Kazdan-Warner problem and the Chern-Simons Higgs problem on compact Riemman surfaces with smooth boundaries.

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

Adams’ inequality with logarithmic weights in ℝ⁴

Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ⁡ ( α | u | 2 1 − β ) d x > + ∞ \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx > + \infty \end{equation*} holds if and only if \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log ⁡ 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log ⁡ e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) W_{0,rad}^{2,2}(B,\omega ) is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of u u in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].

Anisotropic Trudinger–Moser inequalities associated with the exact growth in $${\mathbb {R}}^N$$ and its maximizers

Anisotropic Trudinger–Moser inequalities associated with the exact growth in $${\mathbb {R}}^N$$ and its maximizers

Ground state solutions for Schrödinger–Poisson system with critical exponential growth in [formula omitted

This paper gives weak sufficient conditions to get the existence of ground state solutions for the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) is of critical exponential growth in the sense of Trudinger–Moser inequality. Especially, some new estimates involving the Moser’s sequence of functions are given to get some suitable upper bound for the mountain pass level. Our theorem extends and improves the results of de Figueiredo et al. (1995) and of Chen and Tang (2020).

On a Weighted Adachi–Tanaka Type Trudinger–Moser Inequality in Nonradial Sobolev Spaces

In this article we establish a Trudinger–Moser inequality of Adachi–Tanaka type in nonradial weighted Sobolev spaces for functions defined on the whole \mathbb{R}^{N} . The main tools are the Besecovitch covering lemma and a Trudinger–Moser inequality on the whole space established by S. Adachi and K. Tanaka.

Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

We consider the following two coupled nonlinear Schrödinger system: where the coupling parameter satisfies 0 < λ(x) ≤ λ0 < 1 and the reactions f1, f2 have critical exponential growth in the sense of Trudinger–Moser inequality. Using non‐Nehari manifold method together with the Lions's concentration compactness and the Trudinger‐Moser inequality, we show that the above system has a Nehari‐type ground state solution and a nontrivial solution. Our results improve and extend the previous results.

Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition

Abstract The main aim of this paper is to investigate the existence of nontrivial solutions for a class of fractional Kirchhoff-type problems with right-hand side nonlinearity which is subcritical or critical exponential growth (subcritical polynomial growth) at infinity. However, it need not satisfy the Ambrosetti–Rabinowitz (AR) condition. Some existence results of nontrivial solutions are established via Mountain Pass Theorem combined with the fractional Moser–Trudinger inequality.

Quasilinear Schrödinger equations with singular and vanishing potentials involving nonlinearities with critical exponential growth

In this paper, we study the following class of Schrödinger equations: \[ -\Delta_{N}u+V(|x|)|u|^{N-2}u=Q(|x|)h(u) \quad \text{in } \mathbb{R}^N, \] where $N\geq 2$, $V,Q\colon \mathbb{R}^{N}\rightarrow \mathbb{R}$ are potentials that can be unbounded, decaying or vanishing at infinity and the nonlinearity $h\colon \mathbb{R}\rightarrow \mathbb{R}$ has a critical exponential growth concerning the Trudinger-Moser inequality. By using a variational approach, a version of the Trudinger-Moser inequality and a symmetric criticality type result, we obtain the existence of nonnegative weak and ground state solutions for this class of problems and under suitable assumptions, we obtain a nonexistence result.

Analysis of a fully discrete approximation for the classical Keller–Segel model: Lower and a priori bounds

This paper is devoted to constructing approximate solutions for the classical Keller–Segel model governing chemotaxis. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant.The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds – positivity for the cell density and nonnegativity for the chemoattractant density–, are bounded in the L1(Ω)-norm, satisfy a discrete energy law, and have a priori energy estimates. The latter is achieved by means of a discrete Moser–Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.

Multiple solutions of Choquard equations in ℝ2 with critical exponential growth via penalization method

We deal with the singularly perturbed Choquard equation in the plane with critical exponential growth in the sense of the Trudinger–Moser inequality. Under a local condition imposed on the potential, we show the multiplicity and concentration of positive solutions via penalization technique and variational methods.

New weighted sharp Trudinger–Moser inequalities defined on the whole euclidean space $$ {\mathbb {R}}^N $$ and applications

In this paper, we provide an extension to the whole euclidean space $$ {\mathbb {R}}^N,\ N \ge 2, $$ of the Trudinger–Moser inequalities proved by Calanchi and Ruf (Nonlinear Anal 121:403–411, 2015) involving a logarithmic weight. The inequalities are new and highlight very well the importance of the presence of this type of weight. Next, we prove some version of the concentration-compactness principle due to P.L. Lions giving some new improvements of the Trudinger–Moser inequalities established in the first part of this work. In the light of this last result, we treat some elliptic quasilinear problems involving new type of exponential growth condition at infinity.

Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth

In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: $$\begin{aligned} {\left\{ \begin{array}{ll} -m(\Vert \nabla u\Vert _2^2)\Delta u=f(x,u), \;\;&{} \text{ in } \ \ \Omega ,\\ u= 0, \;\;&{} \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^2$$ is a smooth bounded domain, $$m:\mathbb {R}^+\rightarrow \mathbb {R}^+$$ is a Kirchhoff function, and f has critical exponential growth in the sense of Trudinger–Moser inequality. We develop some new approaches to estimate precisely the minimax level of the energy functional and prove the existence of Nehari-type ground-state solutions and nontrivial solutions for the above problem. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level, and also give a simple proof of a known inequality due to P.L. Lions.

More insights into the Trudinger–Moser inequality with monomial weight

In this paper we present a detailed study of critical embeddings of weighted Sobolev spaces into weighted Orlicz spaces of exponential type for weights of monomial type. More precisely, we give an alternative proof of a recent result by N. Lam [NoDEA 24(4), 2017] showing the optimality of the constant in the Trudinger–Moser inequality. We prove a Poincare inequality for this class of weights. We show that the critical embedding is optimal within the class of Orlicz target spaces. Moreover, we prove that it is not compact, and derive a corresponding version of P.-L. Lions’ principle of concentrated compactness.

Existence of solutions for a fractional Choquard-type equation in $$\mathbb {R}$$ with critical exponential growth

In this paper, we study the following class of fractional Choquard-type equations $$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(I_{\mu }\) is the Riesz potential with \(0<\mu <1\), and F is the primitive function of f. We use variational methods and minimax estimates to study the existence of solutions when f has critical exponential growth in the sense of Trudinger–Moser inequality.