Trudinger Moser Inequality Research Articles

On the Moser–Trudinger inequality in complex space

In this paper we prove the pluricomplex counterpart of the Moser-Trudinger and Sobolev inequalities in complex space. We consider these inequalities for plurisubharmonic functions with finite pluricomplex energy, and we estimate the concerned constants.

Existence of extremals for critical Trudinger-Moser inequalities via the method of energy estimate

We reprove the existence of extremals for the critical Trudinger-Moser inequality on a smooth bounded domain of R2 via the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [11], Mancini-Martinazzi [12] and Mancini-Thizy [13]. For this purpose, unlike [12,13], it suffices to calculate local energy of maximizers for subcritical Trudinger-Moser functionals near the blow-up point.

Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space Rn, including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25], combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

Trudinger–Moser inequalities on harmonic [formula omitted] groups under Lorentz norms

The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on harmonic AN groups S for functions whose gradient is in the Lorentz space L(n,q). Our results show that even if S is not with strictly negative sectional curvature, one can also replace the norm ∫S|∇Su|ndV+∫S|u|ndV by ∫S|∇Su|ndV in the Trudinger–Moser type inequalities.

Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in W1,N(ℝN)

We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] both in the subcritical case [Formula: see text] and critical case [Formula: see text] with [Formula: see text] and [Formula: see text] denotes the surface area of the unit sphere in [Formula: see text]. We will show that MT[Formula: see text] is attained in the subcritical case if [Formula: see text] or [Formula: see text] and [Formula: see text] with [Formula: see text] being the best constant in a Gagliardo–Nirenberg inequality in [Formula: see text]. We also show that MT[Formula: see text] is not attained for [Formula: see text] small which is different from the context of bounded domains. In the critical case, we prove that MT[Formula: see text] is attained for [Formula: see text] small enough. To prove our results, we first establish a lower bound for MT[Formula: see text] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[Formula: see text] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[Formula: see text]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [Formula: see text], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.].

Concentration–Compactness principle for the sharp Adams inequalities in bounded domains and whole space [formula omitted

We prove an improvement of the sharp Adams inequality in W0m,nm(Ω) where Ω is a bounded domain in Rn inspired by Lions Concentration–Compactness principle for the sharp Moser–Trudinger inequality. Our method gives an alternative approach to a Concentration–Compactness principle in W0m,nm(Ω) recently established by do Ó and Macedo. Furthermore, we obtain a sharp threshold for m odd improving the one of do Ó and Macedo. Our approach is also successfully applied to whole space Rn to establish the improvements of the sharp Adams inequalities in Wm,nm(Rn). This type of improvement is still unknown, in general, except the case m=1 due to do Ó, de Souza, de Medeiros and Severo. Our method is a further development of the one of Černy, Cianchi and Hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives.

A sharp Trudinger-Moser type inequality involving Ln norm in the entire space [formula omitted

Let W1,n(Rn) be the standard Sobolev space and ‖⋅‖n be the Ln norm on Rn. We establish a sharp form of the following Trudinger-Moser inequality involving the Ln normsup‖u‖W1,n(Rn)=1∫RnΦ(αn|u|nn−1(1+α‖u‖nn)1n−1)dx<+∞ for any 0≤α<1, where Φ(t)=et−∑n−2j=0tjj!, αn=nωn−11n−1 and ωn−1 is the n−1 dimensional surface measure of the unit ball in Rn. We also show that the above supremum is infinity for all α≥1. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when α>0 is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals.Our results sharpen the recent work [19] in which they show that the above inequality holds in a weaker form when Φ(t) is replaced by a strictly smaller Φ⁎(t)=et−∑n−1j=0tjj! (note that Φ(t)=Φ⁎(t)+tn−1(n−1)!).

Sharp Singular Trudinger–Moser Inequalities Under Different Norms

Abstract The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝ N {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces D N , q ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})} , q ≥ 1 {q\geq 1} , the completion of C 0 ∞ ⁢ ( ℝ N ) {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥ ∇ ⁡ u ∥ N + ∥ u ∥ q {\|\nabla u\|_{N}+\|u\|_{q}} . The case q = N {q=N} (i.e., D N , q ⁢ ( ℝ N ) = W 1 , N ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})} ) has been well studied to date. Our goal is to investigate which type of Trudinger–Moser inequality holds under different norms when q changes. We will study these inequalities under two types of constraint: semi-norm type ∥ ∇ ⁡ u ∥ N ≤ 1 {\|\nabla u\|_{N}\leq 1} and full-norm type ∥ ∇ ⁡ u ∥ N a + ∥ u ∥ q b ≤ 1 {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1} , a &gt; 0 {a&gt;0} , b &gt; 0 {b&gt;0} . We will show that the Trudinger–Moser-type inequalities hold if and only if b ≤ N {b\leq N} . Moreover, the relationship between these inequalities under these two types of constraints will also be investigated. Furthermore, we will also provide versions of exponential type inequalities with exact growth when b &gt; N {b&gt;N} .

On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

On a Hamiltonian System with Critical Exponential Growth

We are interested in finding nontrivial solutions for a Hamiltonian elliptic system in dimension two involving a potential function which can be coercive and nonlinearities that have maximal growth with respect to the Trudinger–Moser inequality. To establish the existence of solutions, we use variational methods combined with Trudinger–Moser type inequalities in Lorentz–Sobolev spaces and a finite-dimensional approximation.

2D Trudinger–Moser inequality for Boltzmann–Poisson equation with continuously distributed multi-intensities

In this paper we study a functional associated with the mean filed limit of the point vortex distribution, that is, $$\begin{aligned} J_{\lambda }(v)=\frac{1}{2}\Vert \nabla v\Vert _2^2-\lambda \int _{I_+}\log \Big (\int _{{\varOmega }} e^{\alpha v} dx \Big ){\mathcal {P}}(d\alpha ), \quad v \in H_0^1({\varOmega }) \end{aligned}$$ where $$\lambda >0$$ is a constant, $${\varOmega }\subset {\mathbb {R}}^2$$ is a smooth bounded domain and $${\mathcal {P}}(d\alpha )$$ is a Borel probability measure on $$I_+=[0, 1]$$ . We show the boundedness of $$J_{\lambda }$$ from below, with borderline inequality in $$\lambda $$ when $${\mathcal {P}}(d\alpha )$$ is continuous and satisfies the suitable assumptions.

Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R2,−Δu+V(x)u=(1|x|μ⁎F(u)|x|β)f(u)|x|β+εh(x)inR2, where V is a positive continuous potential, 0<μ<2, β≥0, 2β+μ≤2, ε is a small parameter and F(s) is the primitive function of f(s). Suppose that the nonlinearity f(s) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

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 estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser–Trudinger inequalities on complete non-compact n-dimensional Riemannian manifolds (n≥2) with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the n-Laplace–Beltrami operator and a critical nonlinearity on n-dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].

Equivalence of Sharp Trudinger-Moser Inequalities in Lorentz-Sobolev Spaces

The critical and subcritical Trudinger-Moser inequalities in Lorentz Sobolev space have been studied by Cassani and Tarsi (Asymptot. Anal. 64(1-2):29–51, 2009), Lu and Tang (Adv. Nonlinear Stud. 16(3):581–601, 2016). In this paper, we will prove that these critical and subcritical Trudinger-Moser inequalities are actually equivalent and thus extend those equivalence results of Lam et al. (Rev. Mat. Iberoam 33(4):1219–1246, 2017) into Lorentz Sobolev spaces.

N-Kirchhoff–Choquard equations with exponential nonlinearity

This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser–Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex–concave problem (Pλ,M) below.

An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

AbstractThe classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$, $u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$ and $s&gt;n/2$. The inequality fails for $s=n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if $n\geqslant 3$ and $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$ is bounded, then $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $L^{p,q}$ is the Lorentz space refinement of $L^{p}$. This inequality fails for $n=2$, and we prove a sharp substitute result: there exists $c&gt;0$ such that for all $\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$ with finite measure, $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$ This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

A Trudinger–Moser inequality for a conical metric in the unit ball

In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$ $(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}\beta}(dx_1^2+\cdots+dx_N^2)$ be a conical metric on $\mathbb{B}$, and $\lambda_p(\mathbb{B})=\inf\left\{\int_\mathbb{B}|\nabla u|^Ndx: u\in W_0^{1,N}(\mathbb{B}),\,\int_\mathbb{B}|u|^pdx=1\right\}$. We prove that for any $\beta\geq 0$ and $\alpha<(1+\frac{p}{N}\beta)^{N-1+\frac{N}{p}}\lambda_p(\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\in W_0^{1,N}(\mathbb{B})$ with $\int_\mathbb{B}|\nabla u|^Ndx-\alpha(\int_\mathbb{B}|u|^p|x|^{p\beta}dx)^{N/p}\leq 1$, there holds $$\int_\mathbb{B}e^{\alpha_N(1+\frac{p}{N}\beta)|u|^{\frac{N}{N-1}}}|x|^{p\beta}dx\leq C,$$ where $|x|^{p\beta}dx=dv_g$, $\alpha_N=N\omega_{N-1}^{1/(N-1)}$, $\omega_{N-1}$ is the area of the unit sphere in $\mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<\beta<0$ and $\alpha=0$ was considered by Adimurthi-Sandeep \cite{A-S}, while the case $p=N=2$, $\beta\geq 0$ and $\alpha=0$ was studied by de Figueiredo-do \'O-dos Santos \cite{F-do-dos}.

Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality in the spirit of Tintarev (J. Funct. Anal. 266, 55–66, 2014) and prove the existence of extremal functions for this improved singular Moser–Trudinger inequality. Our proof is based on the blow-up analysis method. Due to appearance of the singularity of weights, our proof is more complicated and difficult in dealing with analyzing the asymptotic behavior of the sequence of maximizers in the subcritical case near the blow-up point when comparing with the previous works (see, e.g., Nguyen (Ann. Global Anal. Geom. 54(2), 237–256, 2018), Yang (J. Funct. Anal. 239(1), 100–126, 2006)). To overcome these difficulties, we shall prove a classification result for a singular quasi-linear Liouville equation which maybe is of independent interest. Finally, we derive another improvement of the singular Moser–Trudinger inequality in the sprit of Adimurthi and Druet (Comm. Partial Differential Equations 29, 295–322, 2004). Our results extend many well-known Moser–Trudinger type inequalities to more general setting (e.g., singular weights, higher dimension, any domain, etc).

The cone Moser–Trudinger inequalities and applications

In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If $ \beta \in [0,1) $ and $ w_0(x) = |\log |x||^{\beta(n-1)} $ or $ \left( \log \frac{e}{|x|}\right)^{\beta(n-1)} $ then \begin{document}$ \sup\limits_{\int_B | \nabla u|^nw_0 \leq 1 , u \in W_{0,rad}^{1,n}(w_0,B)} \;\; \int_B \exp\left(\alpha |u|^{\frac{n}{(n-1)(1-\beta)}} \;\;\; \right) dx &lt; \infty $\end{document} if and only if $ \alpha \leq \alpha_\beta = n\left[\omega_{n-1}^{\frac{1}{n-1}}(1-\beta) \right]^{\frac{1}{1-\beta}} $ where $ \omega_{n-1} $ denotes the surface measure of the unit sphere in $ \mathbb {R}^n $. The primary goal of this work is to address the issue of existence of extremal function for the above inequality. A non-existence (of extremal function) type result is also discussed, for the usual Moser-Trudinger functional.