Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jungang Li ◽  
Guozhen Lu ◽  
Maochun Zhu
Keyword(s):  
Lower Bound ◽  
Riemannian Manifolds ◽  
Ground State ◽  
Type Inequality ◽  
Concentration Compactness ◽  
Heisenberg Groups ◽  
Stratified Groups ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Noncompact Riemannian Manifolds

Abstract The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

scholarly journals Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenxuan Zheng ◽  
Wenbin Gan ◽  
Shibo Liu
Keyword(s):  
Critical Exponent ◽  
Bounded Domain ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Compactness Principle ◽  
Compactness Principle

AbstractIn this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.

Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝ n and n-Laplace Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 567-585 ◽  
Author(s):  
Caifeng Zhang ◽  
Lu Chen
Keyword(s):  
Positive Constant ◽  
Ground State ◽  
Sufficient Conditions ◽  
Embedding Theorem ◽  
Compact Embedding ◽  
Concentration Compactness ◽  
Critical Nonlinearity ◽  
Concentration Compactness Principle ◽  
Laplace Equations ◽  
Compactness Principle

AbstractIn this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that{W^{1,n}(\mathbb{R}^{n})}is compactly embedded into{L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)}for{p\geq n}and{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the followingn-Laplace equation with critical nonlinearity:($*$){}\left\{\begin{aligned} &\displaystyle{-}\operatorname{div}(|\nabla u|^{n-2}% \nabla u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^{\beta}},\\ &\displaystyle u\in W^{1,n}(\mathbb{R}^{n}),\quad u\geq 0,\end{aligned}\right.where{V(x)\geq c_{0}}for some positive constant{c_{0}}and{f(x,t)}behaves like{\exp(\alpha|t|^{\frac{n}{n-1}})}as{t\rightarrow+\infty}. This work improves substantially related results found in the literature.

scholarly journals Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

2021 ◽  
Author(s):  
Xilin Dou ◽  
xiaoming he
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Concentration Compactness Principle ◽  
Whole Space ◽  
Compactness Principle ◽  
Vanishing Potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

2018 ◽  
Vol 18 (3) ◽  
pp. 429-452 ◽  
Author(s):  
Lu Chen ◽  
Jungang Li ◽  
Guozhen Lu ◽  
Caifeng Zhang
Keyword(s):  
Ground State ◽  
Compact Embedding ◽  
Concentration Compactness ◽  
Critical Nonlinearity ◽  
Ground State Solutions ◽  
Laplacian Equation ◽  
Concentration Compactness Principle ◽  
Formula Group ◽  
Compactness Principle ◽  
Sharp Concentration

AbstractIn this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in{\mathbb{R}^{4}}. We also give a new Sobolev compact embedding which states{W^{2,2}(\mathbb{R}^{4})}is compactly embedded into{L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)}for{p\geq 2}and{0<\beta<4}. As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity:\displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4},where{V(x)}has a positive lower bound and{f(x,t)}behaves like{\exp(\alpha|t|^{2})}as{t\to+\infty}. In the case{\beta=0}, because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming{f(x,t)}and{V(x)}are radial with respect toxand{f(x,t)=o(t)}as{t\rightarrow 0}.

scholarly journals Ground state solutions for periodic Discrete nonlinear Schrödinger equations

2021 ◽  
Vol 6 (12) ◽  
pp. 13057-13071
Author(s):  
Xionghui Xu ◽  
◽  
Jijiang Sun
Keyword(s):  
Ground State ◽  
Linking Theorem ◽  
Concentration Compactness ◽  
Asymptotically Linear ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Concentration Compactness Principle ◽  
Discrete Nonlinear Schrödinger Equations ◽  
Compactness Principle ◽  
Weaker Conditions

<abstract><p>In this paper, we consider the following periodic discrete nonlinear Schrödinger equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} Lu_{n}-\omega u_{n} = g_{n}(u_{n}), \qquad n = (n_{1}, n_{2}, ..., n_{m})\in \mathbb{Z}^{m}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \omega\notin \sigma(L) $(the spectrum of $ L $) and $ g_{n}(s) $ is super or asymptotically linear as $ |s|\to\infty $. Under weaker conditions on $ g_{n} $, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.</p></abstract>

scholarly journals Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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