Trudinger Moser Type 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}.

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.

Existence of normalized solutions for the Chern-Simons-Schrödinger system with critical exponential growth

This paper is dedicated to studying the existence of normalized solutions for a class of Chern-Simons-Schrödinger system, where the nonlinearity possesses critical exponential growth of Trudinger-Moser type. Under some weak assumptions, we obtain several new existence results by employing more delicate estimates and analytical technical. Our results improve and complement the works of Yao et al. (2023) [25] and Yuan et al. (2022) [27].

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 $n=1$.

On a supercritical k-Hessian inequality of Trudinger–Moser type and extremal functions

On a supercritical k-Hessian inequality of Trudinger–Moser type and extremal functions

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 weighted Trudinger–Moser inequalities and applications to some weighted Laplacian equation in with new exponential growth conditions

In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$ Laplacian operator where $1 &lt; q &lt; N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

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.

Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity

AbstractIn this paper, we construct invariant measures and global-in-time solutions for a fractional Schrödinger equation with a Moser–Trudinger type nonlinearity $$\begin{aligned} i\partial _t u= (-\Delta )^{\alpha }u+ 2\beta u e^{\beta |u|^2} \qquad \text{ for }\qquad (x,t)\in \ M\times \mathbb {R}\end{aligned}$$ i ∂ t u = ( - Δ ) α u + 2 β u e β | u | 2 for ( x , t ) ∈ M × R on a compact Riemannian manifold M without boundary of dimension $$d\ge 2$$ d ≥ 2 . To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy (’19) and Sy and Yu (’21). More precisely, we show that if $$s&gt;d/2$$ s &gt; d / 2 or if $$s\le d/2$$ s ≤ d / 2 and $$s\le 1+\alpha $$ s ≤ 1 + α , there exists an invariant measure $$\mu ^{s}$$ μ s and a set $$\Sigma ^s \subset H^s$$ Σ s ⊂ H s containing arbitrarily large data such that $$\mu ^{s}(\Sigma ^s ) =1$$ μ s ( Σ s ) = 1 and that (E) is globally well-posed on $$\Sigma ^{s}$$ Σ s . In the case when $$s&gt;d/2$$ s &gt; d / 2 , we also obtain a logarithmic upper bound on the growth of the $$H^r$$ H r -norm of our solutions for $$r&lt;s$$ r &lt; s . This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert (’21) for the same equation.

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.

A critical elliptic problem involving exponential and singular nonlinearities

We study the existence of a solution to an elliptic problem driven by a fractional p-Laplacian and a nonlinearity which is a product of an exponential and a singular nonlinearity. The exponential singularity is of Trudinger-Moser type and the singularity is of the order \(0<\beta <1\).

Ground state solutions of a Choquard type system with critical exponential growth

In this paper, we study the existence of ground state solutions for the following Choquard type system where and , has exponential critical growth of Trudinger-Moser type, is the primitive function of for i = 1, 2, λ is a continuous and periodic function.

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.