Concentration-compactness Principle Research Articles

The concentration–compactness principle for Orlicz spaces and applications

AbstractIn this paper, we extend the well‐known concentration–compactness principle of P.L. Lions to Orlicz spaces. As an application, we show an existence result to some critical elliptic problem with nonstandard growth.

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Normalized solutions for Sobolev critical fractional Schrödinger equation

Abstract In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: ( − Δ ) s u + λ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R N , ( P m ) ∫ R N ∣ u ∣ 2 d x = m 2 , \hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" voffset="-2.9ex">\left({P}_{m})</mml:mpadded>\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , N ≥ 2 N\ge 2 , m > 0 m\gt 0 , 2 s * ≔ 2 N N − 2 s {2}_{s}^{* }:= \frac{2N}{N-2s} , λ \lambda is an unknown parameter that will appear as a Lagrange multiplier, and f f is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f f , with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to ( P m ) \left({P}_{m}) . We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s = 1 s=1 .

Sign-changing solutions for critical Choquard equation on bounded domain

We consider the existence of sign-changing solutions for the following nonlinear Choquard equation{−Δu=(Iα⁎|u|p)|u|p−2u+λu,x∈Ω,u∈H01(Ω), where N≥3,α∈(0,N),λ∈R, p∈(1,N+αN−2] and Ω⊂RN is a smooth bounded domain. Iα:RN→R is the Riesz potential. In critical case p=N+αN−2, if Ω is symmetric about the axis x1, we firstly develop the limit profiles for the symmetric Palais-Smale sequence by the concentration compactness principle. Then we conclude that the problem admits an odd solution with exactly two nodal domains for λ∈(0,λ1),N≥4andα∈(2,N). In contrast, the local Brezis-Nirenberg type problem −Δu=|u|4N−2u+λu,u∈H01(Ω) can not permit such type of odd symmetry solution.

Existence of nontrivial solutions for a fractional p&q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\&q$\\end{document}-Laplacian equation with sandwich-type and sign-changing nonlinearities

In this paper, we deal with the following fractional p&q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\&q$\\end{document}-Laplacian problem: {(−Δ)psu+(−Δ)qsu=λa(x)|u|θ−2u+μb(x)|u|r−2uinΩ,u(x)=0inRN∖Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left \\{ \ extstyle\\begin{array}{l@{\\quad }l} (-\\Delta )_{p}^{s}u +(-\\Delta )_{q}^{s}u =\\lambda a(x)|u|^{\ heta -2}u+ \\mu b(x)|u|^{r-2}u&\ ext{in}\\;\\ \\Omega , \\\\ u(x)=0 &\ ext{in}\\;\\ \\mathbb{R}^{N}\\setminus \\Omega , \\end{array}\\displaystyle \\right . $$\\end{document} where Ω⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset \\mathbb{R}^{N}$\\end{document} is a bounded domain with smooth boundary, s∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s\\in (0,1)$\\end{document}, (−Δ)ms\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\\Delta )_{m}^{s}$\\end{document}(m∈{p,q})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m\\in \\{p,q\\})$\\end{document} is the fractional m-Laplacian operator, p,q,r,θ∈(1,ps∗]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p,q,r,\ heta \\in (1,p_{s}^{*}]$\\end{document}, ps∗=NpN−sp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p_{s}^{*}=\\frac{Np}{N-sp}$\\end{document}, λ,μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda , \\mu >0$\\end{document}, and the weights a(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a(x)$\\end{document} and b(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b(x)$\\end{document} are possibly sign changing. Using the concentration compactness principle for fractional Sobolev spaces and the Ekeland variational principle, we prove that the problem admits a nonnegative solution for the critical case r=ps∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r=p_{s}^{*}$\\end{document}. Moreover, for the subcritical case r<ps∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r< p_{s}^{*}$\\end{document}, we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem.

Normalized solutions for critical Schrödinger-Poisson system involving the [formula omitted]-subLaplacian in the Heisenberg group

This article deals with existence and multiplicity of normalized solutions for critical Schrödinger-Poisson system involving the p-subLaplacian in the Heisenberg group. Under appropriate assumptions, together with the truncation technique, the concentration-compactness principle, the genus theory, we prove that the existence and multiplicity of the normalized solutions in the Lp-subcritical case. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for the critical p-subLaplacian Schrödinger-Poisson system in the Heisenberg group. Moreover, the results of the paper are completely new even in the Euclidean case.

Existence of stable standing waves for the critical fractional Schrödinger equation with an inhomogeneous combined nonlinearity

We study the orbitally stability of standing waves for the ‐critical inhomogeneous combined source term fractional nonlinear Schrödinger equation: where and . We prove that when or , there exist the standing waves that are orbitally stable by the profile decomposition and concentration compactness principle in .

Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.

Double phase anisotropic variational problems involving critical growth

Abstract In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p ( ⋅ ) p\left(\cdot ) -Laplace equations.

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Abstract The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x &lt; + ∞ , $$\underset{u\in {H}^{1}\left({\mathbb{R}}^{2}\right),\underset{{\mathbb{R}}^{2}}{\int }\left(\vert \nabla u{\vert }^{2}+V\left(x\right){u}^{2}\right)\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\frac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x{&lt; }+\infty ,$$ where 0 &lt; β &lt; 2 $0{&lt; }\beta {&lt; }2$ , V ( x ) ≥ 0 $V\left(x\right)\ge 0$ and may vanish on an open set in R 2 ${\mathbb{R}}^{2}$ . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 ${\mathbb{R}}^{2}$ : (0.2) − Δ u + γ u = f ( u ) | x | β , $$-{\Delta}u+\gamma u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }},$$ where the nonlinearity f $f$ has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${\mathbb{R}}^{n}$ and n $n$ -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 ${\mathbb{R}}^{2}$ : (0.3) − Δ u + V ( x ) u = f ( u ) | x | β . $$-{\Delta}u+V\left(x\right)u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }}.$$

Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.

Critical fractional p-Kirchhoff type problem with a generalized Choquard nonlinearity and magnetic field

In this article, we establish the fractional version of concentration compactness principle to study the existence of solutions for the critical fractional p-Kirchhoff problem with magnetic field, a generalized Choquard nonlinearity, and the Kirchhoff function is non-degenerate or degenerate.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Kirchhoff‐type critical fractional Laplacian system with convolution and magnetic field

AbstractIn this paper, we consider a class of upper critical Kirchhoff‐type fractional Laplacian system with Choquard nonlinearities and magnetic fields. With the help of the limit index theory and the concentration–compactness principles for fractional Sobolev spaces, we establish the existence of infinitely many nontrivial radial solutions for the above system. A distinguished feature of this paper is that the above Kirchhoff‐type system is degenerate, that is, the Kirchhoff term is zero at zero.

Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ωβ(x)=[−lnFo(x)|(n−1)β. This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when β is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality.

Multiple Solutions of a Nonlocal Problem with Nonlinear Boundary Conditions

In this article, we consider a class of nonlocal p(x)-Laplace equations with nonlinear boundary conditions. When the nonlinear boundary involves critical exponents, using the concentration compactness principle, mountain pass lemma, and fountain theorem, we can prove the existence and multiplicity of solutions.

On a critical time-harmonic Maxwell equation in nonlocal media

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media \[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u &amp; \mathrm{in}\ \Omega,\\ \nu\times u=0 &amp; \mathrm{on}\ \partial\Omega, \end{cases} \] where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain, either convex or with $\mathcal {C}^{1,1}$ boundary, $\nu$ is the exterior normal, $\lambda &lt;0$ is a real parameter, $2^{\ast }_{\alpha }=3+\alpha$ with $0&lt;\alpha &lt;3$ is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator $W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$ , we are able to obtain the ground state solutions of the curl–curl equation via the method of constraining Nehari–Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the concentration–compactness principle.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.