Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$.
Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $\exp\left( αt^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.
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.
We prove the existence of positive solutions for a class of semipositone problem with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.
In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.
In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.
In this paper, we are concerned with the critical and subcritical Trudinger-Moser type inequalities for functions in a fractional Sobolev space $H^{1/2,2}$ on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.
In this paper, using the method of blow-up analysis, we obtained a Trudinger-Moser inequality involving L p -norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger-Moser functional. This extends an early result of Yang [27]. Similar result in the Euclidean space was also established by Lu-Yang [17].
We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,α}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.
In this paper, we concern trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. This kind of inequalities were extensively studied by Osgood-Phillips-Sarnak [24], Liu [20], Li-Liu [17], Yang [31, 32] and others. We establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals via the method of blow-up analysis. The results in the current paper generalize those of Li-Liu [17] and Yang [32, 33].
We improve the sharpness of some fractional Moser-Trudinger type inequalities, particularly those studied by Lam-Lu and Martinazzi. As an application, improving upon works of Adimurthi and Lakkis, we prove the existence of weak solutions to the problem $(-Δ)^\frac{n}{2}u=λue^{bu^2} \,\text{ in }Ω,\, 0<λ<λ_1,\,b>0,$ with Dirichlet boundary condition, for any domain $Ω$ in $\mathbb{R}^n$ with finite measure. Here $λ_1$ is the first eigenvalue of $(-Δ)^\frac n2$ on $Ω$.
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.
We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole $\mathbb{R}$ and we give an answer to one of their open questions.
By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends results in the literature for the N-Laplacian operator.
In this paper we prove a Moser-Trudinger inequality for the Euler-Lagrange functional of a general singular Liouville system. We characterize the values of the parameters which yield coercivity for the functional and we give necessary conditions for boundedness from below. We also provide a sharp inequality under some assumptions on the coefficients of the system.
We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.
We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri's inequality for the unit disk in $\mathbb{R}^2$. Moreover we consider Adimurthi-Druet type functionals on compact surfaces with conical singularities and discuss the existence of extremals for such functionals extending previous results by Castò and Roy.
We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.
In this article, we firstly study the cone Moser-Trudinger inequalities and their best exponents $α_2$ on both bounded and unbounded domains $\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study the existence of weak solutions to the nonlinear equation \begin{equation*} \left\{\begin{array}{ll} -Δ_{\mathbb{B}} u=f(x, u), &\mbox{in}\ x\in \mbox{int} (\mathbb{B}), \\ u= 0, &\mbox{on}\ \partial\mathbb{B}, \end{array} \right. \end{equation*} where $Δ_{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_1 =0…
This is a continuation of our previous work [13]. Let $(Σ,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on $(Σ,g)$. Trudinger-Moser inequalities involving $\mathbf{G}$ are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen [7], Iula-Manicini [21], and the authors [13].
We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Apéry's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.
In this paper, we derive a sharp version of the singular Trudinger-Moser inequality, which was originally established by Adimurthi and Sandeep (Nonlinear Differ. Equ. Appl. 2007). Moreover, extremal functions for those singular Trudinger-Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work (J. Differential Equations 2015), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behaviour of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classi…
We study the Dirichlet energy of non-negative radially symmetric critical points $u_μ$ of the Moser-Trudinger inequality on the unit disc in $\mathbb{R}^2$, and prove that it expands as $$4π+\frac{4π}{μ^{4}}+o(μ^{-4})\le \int_{B_1}|\nabla u_μ|^2dx\le 4π+\frac{6π}{μ^{4}}+o(μ^{-4}),\quad \text{as }μ\to\infty,$$ where $μ=u_μ(0)$ is the maximum of $u_μ$. As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of…
Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) …
In this short note, we generalized an energy estimate due to Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to reprove existence of extremals for Trudinger-Moser inequalities of Adimurthi-Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-θ}(r^α\vert u'(r)\vert^βu'(r))'$, where $θ, β\geq 0$ and $α>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Pólya-Szeg{ö} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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 $β\in [0,1)$ and $w_0(x) = |\log |x||^β$ then $$ \sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{αu^{\frac{2}{1-β}}} dx < \infty,$$ if and only if $α\leq α_β= 2\left[2π(1-β) \right]^{\frac{1}{1-β}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $α= α_β$ thereby generalizing the result of Carleson-Chang who proved the case when $β=0$.
We obtain nontrivial solutions of a $(N,q)$-Laplacian problem with a critical Trudinger-Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a …
Druet [6] proved that if $(f_γ)_γ$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_γ)_γ$ solves $$ \begin{cases} &Δu =f_γ(x,u)\,,~~ u>0\text{ in }Ω\,,\\ &u =0\text{ on }\partialΩ\,, \end{cases} $$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_γ$ converges to $4πN$ plus the Dirichlet energy of $u_\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities…
We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -Δu=λue^{u^2}~~&\mbox{in}~Ω,\\[0.5mm] u>0 ~~ &{\text{in}~Ω},\\[0.5mm] u=0~~&\mbox{on}~\partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $λ>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of lo…
The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $Ω\subset\mathbb{R}^n$ is bounded, $ω_{n-1}$ is the $\mathcal{H}^{n-1}$ measure of the unit sphere, then for $α>0$ and $β\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_Ω\frac{e^{α|u|^{n/(n-1)}}}{|x|^β}\leq C \ \Leftrightarrow \ \fracα{α_n}+\fracβ{n}\leq1,\qquad $$ where $α_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(Ω) \ | \ \i…
Combining Carleson-Chang's result with blow-up analysis, we prove existence of extremal functions for certain Trudinger-Moser inequalities in dimension two. This kind of inequality was originally proposed by Adimurthi and O. Druet, extended by the author to high dimensional case and Riemannian surface case, generalized by C. Tintarev to wider cases including singular form and by M. de Souza and J. M. do Ó to the whole Euclidean space R^2. In addition to the Euclidean case, we also consider the Riemannian surface case. The results in the current paper complement that of L. Carleson and A. Chang…
