Moser Trudinger Type Inequalities Research Articles

Concentration of ground state solutions for supercritical zero-mass (N, q)-equations of Choquard reaction

We study the following singularly perturbed (N, q)-equation of Choquard type -εNΔNu-εqΔqu=εμ-N(∫RNK(y)F(u(y))|x-y|μdy)K(x)f(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^N\\Delta _Nu-\\varepsilon ^q\\Delta _qu=\\varepsilon ^{\\mu -N} \\bigg (\\int _{\\mathbb {R}^N}\\frac{K(y)F(u(y))}{|x-y|^{\\mu }}dy\\bigg )K(x)f(u),~x\\in \\mathbb {R}^N, \\end{aligned}$$\\end{document}where Δru=div(|∇u|r-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _ru = \ ext {div}(|\ abla u|^{r-2}\ abla u)$$\\end{document} denotes the usual r-Laplacian operator with r∈{q,N}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\in \\{q,N\\}$$\\end{document} and 1<q<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<q<N$$\\end{document}, ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} is a sufficiently small parameter, K∈C0(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\in C^0(\\mathbb {R}^N)$$\\end{document} satisfies some technical assumptions, 0<μ<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\mu <N$$\\end{document} and F is the primitive of f that fulfills a supercritical exponential growth in the Trudinger–Moser sense. Due to the new version of Trudinger–Moser type inequality introduced in Shen and Rădulescu (Zero-mass (N, q)-Laplacian equation with Stein-Weiss convolution part in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document}: supercritical exponential case. submitted), we aim to derive the existence and concentration of ground state solutions for the given equation using variational method, where the concentrating phenomenon appears at the maximum point set of K as ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+$$\\end{document}.

Nodal solutions for a weighted (N,p)-Kirchhoff-type problem with double exponential growth non-linearity

This paper investigates the existence of sign-changing solutions for a logarithmic weighted Kirchhoff [Formula: see text]-Laplacian problem in the unit ball [Formula: see text] of [Formula: see text] where [Formula: see text]. The non-linearity of the equation is assumed to have double exponential growth, which is motivated by Trudinger–Moser-type inequalities. To prove the existence of these solutions, we employ techniques such as constrained minimization within the Nehari set, a quantitative deformation lemma and results from degree theory.

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

Nonlinear divergence equation with potentials vanishing in some direction at infinity: the exponential critical case

The purpose of this article is to state some weighted Sobolev embedding involving functions that vanishing only in a direction. In this setting we prove a weighted Trudinger–Moser type inequality and as an application, we addressed the existence of solutions to a class of elliptic equation of the form −div(a(x)∇u)+V(x)u=K(x)f(u)inR2, where the nonlinearity f has exponential critical growth in sense of Trudinger–Moser.

Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n&gt;1$), while they have the same singularities of $\phi$ when $n=1$.

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

A Sharp Moser–Trudinger Type Inequality Involving Adimurthi–Druet Term in Two Dimensional Hyperbolic Space

Least energy nodal solutions for a weighted (N,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(N, p)$\\end{document}-Schrödinger problem involving a continuous potential under exponential growth nonlinearity

This article aims to investigate the existence of nontrivial solutions with minimal energy for a logarithmic weighted (N,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(N,p)$\\end{document}-Laplacian problem in the unit ball B of RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{N}$\\end{document}, N>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N>2$\\end{document}. The nonlinearities of the equation are critical or subcritical growth, which is motivated by weighted Trudinger–Moser type inequalities. Our approach is based on constrained minimization within the Nehari set, the quantitative deformation lemma, and degree theory results.

Extremal functions for the critical Trudinger–Moser inequality with logarithmic Kernels

In this paper we study Moser–Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we sharpen the results in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935] under critical growth assumptions and give answers to some questions left open in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935].

Sign-changing solutions for a weighted Schrödinger–Kirchhoff equation with double exponential nonlinearities growth

This work is devoted to the existence of least energy nodal solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball [Formula: see text] of [Formula: see text]. The nonlinearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. By using the constrained minimization in Nehari set, the quantitative deformation Lemma and degree theory results, we prove our existence result.

A quantization proof of the uniform Yau–Tian–Donaldson conjecture

Using quantization techniques, we show that the \delta -invariant of Fujita–Odaka coincides with the optimal exponent in a certain Moser–Trudinger type inequality. Consequently, we obtain a uniform Yau–Tian–Donaldson theorem for the existence of twisted Kähler–Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger–Colding–Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

On a sharp inequality of Adimurthi–Druet type and extremal functions

Our main purpose is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi–Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger–Moser type inequality in de Oliveira and do Ó (Calc Var Part Differ Equ 52:125–163, 2015) for the entire space. In addition, we perform the two-step strategy due to L. Carleson and S-Y.A. Chang together with blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under a subcritical regime for some special cases.

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

On the global existence of solutions to chemotaxis system for two populations in dimension two

We consider the global existence for the following fully parabolic chemotaxis system with two populations \[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, &amp; x\in\Omega,\ t&gt;0, \\ v_t=\Delta v-v+u_1+u_2, &amp; x\in\Omega,\ t&gt;0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), &amp; x\in\Omega, \end{array} \right. \]where $\Omega =\mathbb {R}^2$ or $\Omega =B_R(0)\subset \mathbb {R}^2$ supplemented with homogeneous Neumann boundary conditions, $\kappa _i,\chi _i&gt;0,$$i=1,2$. The global existence remains open for the fully parabolic case as far as the author knows, while the existence of global solution was known for the parabolic-elliptic reduction with the second equation replaced by $0=\Delta v-v+u_1+u_2$ or $0=\Delta v+u_1+u_2$. In this paper, we prove that there exists a global solution if the initial masses satisfy the certain sub-criticality condition. The proof is based on a version of the Moser–Trudinger type inequality for system in two dimensions.

A Trudinger–Moser type inequality in a strip and applications

This paper deals with an improvement of the Trudinger–Moser inequality for the Sobolev space W1,N(Ω) involving domains of the type Ω=RN−1×(a,b), where N≥2 and a<b. As an application of this result and by using minimax methods, we establish sufficient conditions for the existence of solutions for a class of elliptic problems with Neumann boundary conditions and nonlinearities with critical exponential growth.

Nonlinear Schrödinger equations involving exponential critical growth with unbounded or vanishing potentials

In this paper we deal with the following class of nonlinear Schrödinger equations − Δ u + V ( | x | ) u = λ Q ( | x | ) f ( u ) , x ∈ R 2 , where λ &gt; 0 is a real parameter, the potential V and the weight Q are radial, which can be singular at the origin, unbounded or decaying at infinity and the nonlinearity f ( s ) behaves like e α s 2 at infinity. By performing a variational approach based on a weighted Trudinger–Moser type inequality proved here, we obtain some existence and multiplicity results.

On a quasilinear logarithmic N-dimensional equation involving exponential growth

In this paper, we investigate a quasilinear logarithmic N-dimensional equation with radial potentials which can be singular at the origin, unbounded or decaying at infinity and a nonlinearity behaves like eα|s|N/(N−1) at infinity. The key point of our approach is a new weighted Trudinger-Moser type inequality.

Trudinger–Moser type inequalities with a symmetric conical metric and a symmetric potential

Let (M,g) be a closed Riemann surface, where the metric g has certain conical singularities at finite points. Suppose Γ is a group with elements of isometries acting on (M,g). In this paper, Trudinger–Moser inequalities involving Γ and the operador Δg+V are established, where Δg denotes the Laplace–Beltrami operator associated to g and the potential V:M→(0,∞) belongs to a class of symmetric and continuous functions. Moreover, via the method of blow-up analysis, the corresponding extremals are also obtained.

Gagliardo-Nirenberg type inequalities on Lorentz, Marcinkiewicz and weak-𝐿^{∞} spaces

We establish the Gagliardo-Nirenberg inequality, Trudinger-Moser inequality and John-Nirenberg inequality using the Lorentz spaces L p , α L^{p,\alpha } , the Marcinkiewicz space L q , ∞ L^{q,\infty } and the weak- L ∞ L^{\infty } space W W introduced by Bennett, DeVore and Sharpley [Ann. of Math. (2) 113 (1981), pp. 601–611]. As consequences, we obtain the Gagliardo-Nirenberg type inequality with weak- L ∞ L^{\infty } norm and BMO norm, Trudinger-Moser type inequality and John-Nirenberg type estimate with B M O BMO norm and weak- L 1 L^{1} norm.

Critical and subcritical Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Though the sharp Trudinger-Moser inequalities on compact Riemannian manifolds have been known for some times, optimal constants for such inequalities on complete noncompact Riemannian manifolds are still unknown except in some special cases (such as hyperbolic spaces or Hadamard manifolds). In this paper, we will establish the sharp critical and subcritical Trudinger-Moser type inequalities with best constants on any complete and noncompact Riemannian manifold (M,g) whose Ricci curvature and its injectivity radius have lower bounds. (See Theorems 1.3 and 1.4.)More precisely, we will establish the following sharp critical Trudinger-Moser inequality on such manifolds (M,g,dVg):supu∈W1,n(M),||u||1,τ≤1⁡∫Mϕ(αn|u|nn−1)dVg≤C(n,τ), where τ>0, ϕ(t)=∑k=n−1∞tkk!, αn=nωn−11n−1, and ωn−1 is the area of the unit sphere in Rn, ||u||1,τ=(∫Mτ|u|n+|∇u|ndVg)1n. Moreover, we also prove that the following sharp subcritical Trudinger-Moser inequalitysupu∈W1,n(M),||∇u||n≤1⁡1||u||Ln(M)n∫Mϕ(α|u|nn−1)dVg≤C(n,α) in the sense that the supremum is finite for all α<αn, but infinite for α≥αn.The proofs of both the critical and subcritical Trudinger-Moser inequalities rely crucially on the Trudinger-Moser inequality on any compact manifold M with or without boundary, which is of its own interest. The important property we establish in this paper is that the upper bound C(n,M) for the Trudinger-Moser supremum is continuous and monotone increasing with respect to the volume of M. (See Theorems 1.1 and 1.2.) This upper bound property allows us to carry out a symmetrization-free argument from sharp local inequalities on compact manifolds to the optimal global inequalities on complete and noncompact Riemannian manifolds. Our optimal results sharpen substantially the existing ones on such Riemannian manifolds in the literature.

On a zero-mass [formula omitted]-Laplacian equation in [formula omitted] with exponential critical growth

In this work, we investigate the existence of solution for the quasilinear elliptic equation −ΔNu−Δqu=f(u)inRN,where 1<q<N and the nonlinearity f has exponential critical growth in the Trudinger–Moser sense. In order to obtain the solution, we use a variational approach based on a new Trudinger–Moser type inequality which is proved here.