We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.
We give a quantization analysis to an elliptic system (Gelfand-Liouville type system) with Dirichlet condition. An application, we have a com-pactness result for an elliptic system with Lipschitz condition.
In the paper we show that critical sequences associated with the mountain pass level for semilinear elliptic problems on $\R^N$ converge when the non-linearity is subcritical, superlinear and satisfies the penalty condition $F_\infty(s)<F(x,s)$. The proof suggests a concentration compactness framework for minimax problems, similar to that of P.-L.Lions for constrained minima.
This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow to quantify the difference in compactness. The definition of MCFs involves a Fouri…
The concentration compactness framework for semilinear elliptic equations without compactness, set originally by P.-L.Lions for constrained minimization in the case of homogeneous nonlinearity, is extended here to the case of general nonlinearities in the standard mountain pass setting of Ambrosetti-Rabinowitz. In these setting, existence of solutions at the mountain pass level is verified under a single penalty condition analogous to that in the Lions' case. Problems on the whole space and problems with critical nonlinearity are considered. Particular attention is given to nonhomogeneous crit…
In this paper we extend the well-known concentration -- compactness principle of P.L. Lions to the variable exponent case. We also give some applications to the existence problem for the $p(x)-$Laplacian with critical growth.
In this paper we use a concentration and compactness argument to prove the existence of a nontrivial nonradial solution to the nonlinear Schrodinger-Poisson equations in R3, assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions
For $p \in (1,N)$ and $Ω\subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$ is the completion of $C_c^{\infty}(Ω)$ with respect to the norm $\left( \int_Ω|\nabla u|^p \right)^ \frac{1}{p}.$ Using the $p$-capacity, we define a norm and then identify the Banach function space $\mathcal{H}(Ω)$ with the set of all $g$ in $L^1_{loc}(Ω)$ that admits the following Hardy-Sobolev type inequality: \begin{eqnarray*} \int_Ω |g| |u|^p \leq C \int_Ω |\nabla u|^p, \forall\; u \in \mathcal{D}^{1,p}_0(Ω), \end{eqnarray*} for some $C>0.$ Further, we characterize the set of all $g$ in $\m…
This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration--compactness method.
In this paper, we give an overview of the authors' work on applications of the method of concentration-compactness to global well-posedness, scattering, blow-up and universal profiles for the energy critical wave equation in the non-radial setting. New results and proofs are also given.
We study a variational problem for the first Schrodinger eigenvalue on closed Riemannian surfaces. More precisely, we explore concentration-compactness properties of sequences formed by its extremal potentials.
We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.
In this article we develop a concentration compactness type principle in a variable exponent setup. As an application of this principle we discuss a problem involving fractional `{\it $(p(x),p^+)$-Laplacian}' and power nonlinearities with exponents $(p^+)^*$, $p_s^*(x)$ with the assumption that the critical set $\{x\inΩ:p_s^*(x)=(p^+)^*\}$ is nonempty.
In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.
We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in term…
We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1+4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gerard profile decomposition [1] and a concentration compactness/rigidity argument by Kenig-Merle [10], following the method developed by the first author and Schlag [20] in the context of critical wave maps.
The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space $\mathcal{D}^{s,2} (\mathbb{R}^N)$ for $0<s<\min\{1,N/2\}.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional scalar field equation $(-Δ)^{s} u = f(x,u)$ for $0<s<1.$ Moreover, using an analytic framework based on $\mathcal{D}^{s,2}(\mathbb{R}^N),$ we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.
In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, in the context of fractional Sobolev spaces with variable exponents, especially for nonlinear systems. As an application of the results, we show the existence and assymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with exponent variables and critical growth conditions in $\mathbb{R}^{N}$.
We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation- and dilation-invariant case. They also give in particular interpolation estimates between Sobolev spaces and functions of bounded mean oscillation. The proofs rely on Sobolev integral representation formulae and maximal function theory. Fractional Sobolev spaces are also covered.
We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.
The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schrödinger equation $(-Δ)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the…
We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubin's theorem for functions on $\mathbb{S}^{n}$ with zero first order moments of the area element to higher order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher order Sobolev spaces.
By means of the concentrated compactness method of Bahouri-Gerard and Kenig-Merle, we prove global existence and regularity for wave maps with smooth data and large energy from 2+1 dimensions into the hyperbolic plane. The argument yields an apriori bound of the Coulomb gauged derivative components of our wave map relative to a suitable norm (which holds the solution) in terms of the energy alone. As a by-product of our argument, we obtain a phase-space decomposition of the gauged derivative components analogous to the one of Bahouri-Gerard.
Super-Liouville equation with a Dirichlet-chiral boundary
Central China Normal University（华中师范大学）；School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis and Applications (Ministry of Education), Hubei Key Laboratory of Mathematical Sciences, Central China Normal University（华中师范大学数学与统计学院，非线性分析与应用教育部重点实验室，湖北省数学科学重点实验室）
In this paper we consider the nonlinear Choquard equation $$ -Δu+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^μ}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<μ<N$, $N\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \cite{BC1…
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 …
We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere $\mathbb{S}^m$, $m \geq 1$, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial w…
Energy-Casimir functionals are a useful tool for the construction of steady states and the analysis of their nonlinear stability properties for a variety of conservative systems in mathematical physics. Recently, Y. Guo and the author employed them to construct stable steady states for the Vlasov-Poisson system in stellar dynamics, where the energy-Casimir functionals act on number density functions on phase space. In the present paper we construct natural, reduced functionals which act on mass densities on space and study compactness properties and the existence of minimizers in this context.…
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
We study blow-up and quantization phenomena for a sequence of solutions $(u_k)$ to the prescribed $Q$-curvature problem $$ (-Δ)^nu_k= Q_ke^{2nu_k}\quad \text{in }Ω\subset\mathbb{R}^{2n},\quad \int_Ωe^{2nu_k}dx\leq C,$$ under natural assumptions on $Q_k$. It is well-known that, up to a subsequence, either $(u_k)$ is bounded in a suitable norm, or there exists $β_k\to\infty$ such that $ u_k=β_k(φ+o(1))$ in $Ω\setminus (S_1\cup S_φ)$ for some non-trivial non-positive $n$-harmonic function $φ$ and for a finite set $S_1$, where $S_φ$ is the zero set of $φ$. We prove quantization of the total curvat…
In this work we use variational methods to prove results on existence and concentration of solutions to a problem in $\mathbb{R}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV(\mathbb{R}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural "relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being m…
Concentration-compactness is used to prove compactness of maximising sequences for a variational problem governing symmetric steady vortex-pairs in a uniform planar ideal fluid flow, where the kinetic energy is to be maximised and the constraint set comprises the set of all equimeasurable rearrangements of a given function (representing vorticity) that have prescribed impulse (lnear momentum). A form of orbital stability is deduced.
We prove that the family $\mathcal{F}_C(D)$ of all meromorphic functions $f$ on a domain $D\subseteq \mathbb{C}$ with the property that the spherical area of the image domain $f(D)$ is uniformly bounded by $C π$ is quasi--normal of order $\le C$. We also discuss the close relations between this result and the well--known work of Brézis and Merle on blow--up solutions of Liouville's equation. These results are completely in the spirit of Gromov's compactness theorem, as pointed out at the end of the paper.
Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that an…
This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H…
We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family $f_λ(x,v)\in L^p$ satisfying some appropriate transport relation $$v\cdot\nabla_x f_λ= (1-Δ_x)^\fracβ{2}(1-Δ_v)^\fracα{2}g_λ$$ may be inferred solely from its compactness in $v$. This method is introduced as an alternative to the lack of known suitable averaging lemmas in $L^1$ when the right-hand side of the transport equation has very low regularity, due to an e…
In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.
Large mass limits of $\mathrm{G}_2$ and Calabi--Yau monopoles: calibrated concentration, Higgs zeros, and abelianization
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院数学系数学、几何与动态系统中心）
Small-energy regularity for stationary weak solutions of the Liouville equation
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics, Yunnan Normal University（云南师范大学数学学院）
Concentration Phenomena of Normalized Solutions of Critical Biharmonic Equations with Combined Nonlinearities in $\mathbb{R}^{N}$
College of Mathematics, Changchun Normal University（长春师范大学数学学院）；Faculty of Education, University of Ljubljana（卢布尔雅那大学教育学院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics, University of Ljubljana（卢布尔雅那大学数学、物理与力学研究所）
Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.
