Moser Trudinger Type Inequalities Research Articles

A sharp Trudinger-Moser type inequality in $\mathbb {R}^2$

In this paper, we establish a sharp Trudinger-Moser type inequality for a class of Schrödinger operators in $\mathbb {R}^2$. We obtain a result related to the compactness of the embedding of a subspace of $W^{1,2}(\mathbb {R}^2)$ into the Orlicz space $L_{\phi }(\mathbb {R}^2)$ determined by $\phi (t)=e^{\beta t^{2}}-1$. Our result is similar to one obtained by Adimurthi and Druet for smooth bounded domains in $\mathbb {R}^2$, which is closely related to a compactness result proved by Lions. Furthermore, similarly to what has been done by Carleson and Chang, we prove the existence of an extremal function for this Trudinger-Moser inequality by performing a blow-up analysis. Trudinger-Moser type inequalities have a wide variety of applications to the study of nonlinear elliptic partial differential equations involving the limiting case of Sobolev inequalities and have received considerable attention in recent years.

A Trudinger–Moser inequality in a weighted Sobolev space and applications

We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation urn:x-wiley:dummy:mana201200315:equation:mana201200315-math-0001where , f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.

Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality

We prove a concentration-compactness principle for the Trudinger-Moser functional associated with a class of weighted Sobolev spaces including fractional dimensions. Based in this result and using blow up analysis we establish a sharp form of Trudinger-Moser type inequality for this class of weighted Sobolev spaces. Moreover, we discuss the existence of extremal for the maximizing problem associated with this new Trudinger-Moser inequality.

An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean-field equation of the equilibrium turbulence with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing so, we present a new Moser–Trudinger-type inequality under suitable conditions on the centre of mass and the scale of concentration of both eu and e−u, where u is the unknown function in the equation.

A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold X. This functional can be seen as a generalization of Mabuchiʼs K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tianʼs α-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.

Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring

We prove that for radially symmetric functions in a ring $\Omega = ${$ x \in \mathbb{R}^n, n \geq 2 : r \leq |x| \leq R $} a special type of Trudinger-Moser-like inequality holds. Next we show how to infer from it a lack of blowup of radially symmetric solutions to a Keller-Segel system in $\Omega$.

On Singular Trudinger–Moser Type Inequalities for Unbounded Domains and Their Best Exponents

Generalizations of the Trudinger–Moser inequality to Sobolev spaces with singular weights are considered for any smooth domain Ω ⊂ ℝ N . Furthermore, we show that the resulting inequalities are sharp obtaining the best exponents.

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space \(H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})\) with n∈ℕ and 1<q≤p<∞, which coincides with the usual critical Sobolev space \(H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})\) in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant Cp,q depending only on p and q such that $$ \|f\|_{{\mathcal{M}}_{r,\frac{q}{p}r}} \leq C_{p,q}r\|f \|_{{\mathcal{M}}_{p,q}}^{\frac{p}{r}}\bigl\|(-\Delta)^{\frac{n}{2p}} f\bigr\|_{{\mathcal{M}}_{p,q}}^{1-\frac{p}{r}} $$ (GN) for all \(u\in H\mathcal{M}^{\frac{n}{p}}_{p,q}( \mathbb{R}^{n})\) and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space \(H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})\), the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765–769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531–540, 1991) and Ozawa (J. Func. Anal. 127:259–269, 1995) with the growth order \(r^{1-\frac{1}{p}}\) as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brezis-Gallouet-Wainger type inequality in the critical Sobolev-Morrey space.

Scattering for the two-dimensional NLS with exponential nonlinearity

We investigate existence and asymptotic completeness of the wave operators for nonlinear Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger–Moser type inequality. We prove that if the Hamiltonian is below the critical value, then the solution approaches a free Schrödinger solution at the time infinity.

Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space

In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger–Moser type inequality. Under some assumptions on the potential and the nonlinearity, it is proved that there is a nontrivial positive weak solution to this equation. Also it is shown that there are two distinct positive weak solutions to a perturbation of the equation. The method of proving these results is combining Trudinger–Moser type inequality, Mountain-pass theorem and Ekelandʼs variational principle.

Exponential problem on a compact Riemannian manifold without boundary

Let ( M , g ) be an n -dimensional compact Riemannian manifold without boundary. A Trudinger–Moser-type inequality says that sup ‖ u ‖ W 1 , n ≤ 1 ∫ M e α n | u | n n − 1 d v g < ∞ , where ‖ u ‖ W 1 , n is the usual Sobolev norm of u ∈ W 1 , n ( M ) , α n = n ω n − 1 1 n − 1 , and ω n − 1 is the area of the unit sphere S n − 1 . Using this inequality, when ε > 0 is small enough, we establish sufficient conditions under which the quasilinear equation − Δ n u + | u | n − 2 u = f ( x , u ) + ε h ( x ) has at least two nontrivial weak solutions in W 1 , n ( M ) , where − Δ n u = − div g ( | ∇ u | n − 2 ∇ u ) , f ( x , u ) behaves like e γ | u | n n − 1 as | u | → ∞ for some γ > 0 , and h ≢ 0 belongs to the dual space of W 1 , n ( M ) .

Adams type inequalities and related elliptic partial differential equations in dimension four

Motivated by Ruf–Saniʼs recent work, we prove an Adams type inequality and a singular Adams type inequality in the whole four-dimensional Euclidean space. As applications of those inequalities, a class of elliptic partial differential equations are considered. Existence of nontrivial weak solutions and multiplicity results are obtained via the mountain-pass theorem and the Ekelandʼs variational principle. This is a continuation of our previous work about singular Trudinger–Moser type inequality.

SEVERAL ANALYTIC INEQUALITIES IN SOME Q−SPACES

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n})$. Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n})$.

Energy Properness and Sasakian-Einstein Metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.

SEVERAL ANALYTIC INEQUALITIES IN SOME Q−SPACES

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$

Uniqueness and symmetry of equilibria in a chemotaxis model

We consider in a disc of a class of parameter-dependent, nonlocal elliptic boundary value problems that describes the steady states of some chemotaxis systems. If the appearing parameter is less than an explicit critical value, we establish several uniqueness results for solutions that are invariant under a group of rotations. Furthermore, we discuss the associated consequences for the time asymptotic behavior of the solutions to the corresponding time dependent chemotaxis systems. Our results also provide optimal constants in some Moser–Trudinger type inequalities.

Elliptic equations and systems with critical Trudinger-Moser nonlinearities

In this article we give first a survey on recent results on some Trudinger-Moser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sense of Trudinger-Moser. Furthermore, recent results concerning systems of such equations will be discussed.

Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in $${\mathbb{R}^N}$$

We discuss the existence of a maximizer for a maximizing problem associated with the Trudinger–Moser type inequality in \({\mathbb{R}^N(N\geq 2)}\). Different from the bounded domain case, we obtain both of the existence and the nonexistence results. The proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences.

Global well-posedness and linearization of a semilinear 2D wave equation with exponential type nonlinearity

Abstract We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard [J. Funct. Anal. 141: 60–98, 1996]. The proof uses Moser–Trudinger type inequalities and the energy estimate.