Trudinger Inequality Research Articles

Singular Trudinger–Moser inequality with Lp norm and the existence of extremals

In this paper, we establish the singular Trudinger–Moser inequality with norm in a bounded domain and the existence of extremal functions by blow-up analysis. As our first main result, we prove that for , , and , the singular Trudinger–Moser inequality holds, where is the first eigenvalue . Furthermore, we prove the existence of extremals for the singular Trudinger–Moser inequality with norm. Our results improve that of J. Zhu (Zhu J. Improved Moser–Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud 2014;14:273–293) into the singular case.

Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

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.

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

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.

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.

Extremal functions for sharp Moser–Trudinger type inequalities in the whole space [formula omitted

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space RN, N≥2 with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser–Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser–Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser–Trudinger inequality and singular Moser–Trudinger inequality in whole space RN due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.

Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

Improved Adams-type inequalities and their extremals in dimension 2m

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

A Priori Estimate for the Complex Monge–Ampère Equation

In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampere equation, arXiv:2003.06056v1 (2020)) to establish the uniform estimate and the Holder continuity for solutions to the complex Monge–Ampere equation with the right-hand side in $$L^p$$ for any given $$p>1$$ . Our proof uses various PDE techniques but not the pluripotential theory.

The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints

This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case $$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$where $$a>0$$, $$\beta \in [0,N)$$, $$\Phi _N(t) = e^t -\sum _{k=0}^{N-2} \frac{t^k}{k!}$$, $$\alpha _N = N \omega _{N-1}^{1/(N-1)}$$, and $$\omega _{N-1}$$ denotes the surface area of the unit sphere in $${\mathbb {R}}^N$$. More precisely, we study the effect of the parameter a to the attainability of $$MT_{N}(a,\beta )$$. We will prove that for each $$\beta \in [0,N)$$ there exist the thresholds $$a_*(\beta )$$ and $$a^*(\beta )$$ such that $$MT_{N}(a,\beta )$$ is attained for any $$a \in (a_*(\beta ), a^*(\beta ))$$ and is not attained for $$a a^*(\beta )$$. We also give some qualitative estimates for $$a_*(\beta )$$ and $$a^*(\beta )$$. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do O, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).

Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications

The main aim of this article is to study non-singular version of Moser–Trudinger and Adams–Moser–Trudinger inequalities and the singular version of Moser–Trudinger equality in the Cartesian product of Sobolev spaces. As an application of these inequalities, we study a system of Kirchhoff equations with exponential non-linearity of Choquard type.

Extremals for Fractional Moser–Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates

Abstract We prove the existence of extremals for fractional Moser–Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler–Lagrange equation, which requires new sharp estimates obtained via commutator techniques.

Supercritical Moser–Trudinger inequalities and related elliptic problems

Given $\alpha >0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (\alpha_n + |x|^\alpha) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( \alpha_n |u|^{\frac{n}{n-1} + |x|^\alpha} \big) dx < +\infty, \] where $W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the class of functions $W^{1,n}_{0,{\rm rad}}(B)$, we should emphasize that the above inequalities fail in $W^{1,n}_{0,{\rm rad}}(B)$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.

Sharp Adams–Moser–Trudinger type inequalities in the hyperbolic space

The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space \mathbb H^n . First, we prove a sharp Adams’ inequality of order two with the exact growth condition in \mathbb H^n . Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of \mathbb H^n , which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in \mathbb H^n . Our proofs rely on the symmetrization method extended to hyperbolic spaces.

A double mean field equation related to a curvature prescription problem

We study a double mean field–type PDE related to a prescribed curvature problem on compacts surfaces with boundary:(0.1){−Δu=2ρ(Keu∫ΣKeu−1|Σ|)in Σ,∂νu=2ρ′(heu2∫∂Σheu2−1|∂Σ|)on ∂Σ.Here ρ and ρ′ are real parameters, K,h are smooth positive functions on Σ and ∂Σ respectively and ν is the outward unit normal vector to ∂Σ.We provide a general blow–up analysis, then a Moser–Trudinger inequality, which gives energy–minimizing solutions for some range of parameters. Finally, we provide existence of min–max solutions for a wider range of parameters, which is dense in the plane if Σ is not simply connected.

Fractional Adams–Moser–Trudinger type inequality on Heisenberg group

The aim of this paper is to establish singular fractional Adams–Moser–Trudinger inequality for both bounded and unbounded domains in the Heisenberg group Hn. We first establish fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.12) and then using this inequality and Hardy–Littlewood–Sobolev inequality adapted to the result of R.O’Neil(1963), we establish singular fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.13). We also establish singular fractional Adams–Moser–Trudinger type inequality in Hn, using the approach of N.Lam and G.Lu (2012, 2013). The main idea of our approach is that any function in higher order fractional Sobolev space in Heisenberg group can be represented in terms of Riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator, we establish fractional Adams–Moser–Trudinger type inequality in Hn. In this paper, our approach is quite simple and free from symmetrization arguments. As an applications of our theorems, we establish the existence of solution to the following class of problems Tαu=f(ξ,u)|ξ|a+b(ξ)|u|γ−1uinΩ,u=0inHn∖Ω,where Ω is a bounded subset of Hn of class C0,1 with bounded boundary and 0≤a<Q,f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small L2-perturbation, that is, there exists a small η>0 with 0<‖b‖L2(Ω)<η,0≤γ<1 and α=Q2.

The Hardy–Moser–Trudinger inequality via the transplantation of Green functions

<p style='text-indent:20px;'>We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, <i>A Hardy–Moser–Trudinger inequality</i>, Adv. Math, <b>230</b> (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

The weighted Moser–Trudinger inequalities of Adimurthi–Druet type in [formula omitted

Let N≥2,γ≤β<N and λ∈[0,1), we show that sup‖∇u‖NN+‖u‖N,γN≤1∫RNΦN(α(1+λ‖u‖N,γN)1N−1|u|NN−1)|x|−βdx<∞where ‖u‖N,γN=∫RN|u|N|x|−γdx and ΦN(t)=et−∑i=0N−2tii! if and only if α≤α¯N,β=(1−βN)αNif β∈[0,N)αN∕(1−β∕N)1∕(N−1)if β<0, with αN=N(2πN∕2∕Γ(N∕2))1∕(N−1). This value of α¯N,β is new for β<0 or γ<0≤β. Especially, for 0≤γ≤β<N, we obtain an improved singular Moser–Trudinger inequality in the spirit of Adimurthi–Druet. We also study the attainability of the supremum above for α=α¯N,β.

Adams inequalities for Riesz subcritical potentials

We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results in Fontana and Morpurgo (2011). The integral operators involved, which we call “Riesz subcritical”, have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on Rn, where the kernel is large, but they behave better where the kernel is small. The new element is a “critical integrability” condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name “Riesz subcritical domains”. Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser–Trudinger inequalities on Rn, on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincaré inequality holds.