Concentration-compactness Principle Research Articles

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 < + ∞ , $$\underset{u\in {H}^{1}({\mathbb{R}}^{2}),\underset{{\mathbb{R}}^{2}}{\int }(\vert \nabla u{\vert }^{2}+V(x){u}^{2})\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\tfrac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x< +\infty ,$$ where 0 < β < 2 $0< \beta < 2$ , V ( x ) ≥ 0 $V(x)\ge 0$ 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 , $${-}{\Delta }u+\gamma u=\frac{f(u)}{\vert x{\vert }^{\beta }},$$ 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 . $${-}{\Delta }u+V(x)u=\frac{f(u)}{\vert x{\vert }^{\beta }}.$$

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.

Generalized noncooperative Schrödinger–Kirchhoff–type systems in RN$\mathbb {R}^N$

AbstractWe consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compactness principle for weighted‐variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.

Ground state solution for fractional p-Choquard equations with upper critical exponent

In this paper, we investigate the following fractional p-Choquard equations with upper critical exponent involving two potentials:(−Δ)psu+V(x)|u|p−2u=[Iα⁎|u|ps,α⁎]|u|ps,α⁎−2u+K(x)f(x,u),x∈RN, where N⩾1, 0<s<1, 2⩽p<Ns, ps,α⁎=p(N+α)2(N−ps), Iα is the Riesz potential of order α∈(0,1), the potentials V,K:RN→R and nonlinearity f:RN×R→R are continuous functions satisfying some natural hypotheses. We show the existence of a ground state solution to above equations when V and f are periodic and asymptotically periodic in x, respectively, by using the idea of Nehari manifold technique and the Lions' concentration compactness principle.

Existence of solutions to a critical Kirchhoff equation with logarithmic perturbation

In this paper, the following Kirchhoff type elliptic equation − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u ln ⁡ | u | 2 + | u | 4 u , which involves a power type nonlinearity with critical Sobolev exponent and a logarithmic type perturbation, is investigated. The fact that the logarithmic term satisfies neither the monotonicity condition nor the Ambrosetti-Rabinowitz condition brings some additional difficulties when one is looking for weak solutions to this problem. By the use of the Mountain Pass Lemma and the concentration compactness principle, it is proved that the problem admits a ground state solution for q ∈ ( 4 , 6 ) and for any 0 $ ]]> λ > 0 . It is worth pointing out that the analysis includes both the nondegenerate case (a>0) and the degenerate case (a = 0).

The Existence Result for a p -Kirchhoff-Type Problem Involving Critical Sobolev Exponent

In this paper, by using the mountain pass theorem and the concentration compactness principle, we prove the existence of a positive solution for a p -Kirchhoff-type problem with critical Sobolev exponent.

Multiplicity results for generalized quasilinear critical Schrödinger equations 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}

Multiplicity results are proved for solutions both with positive and negative energy, as well as nonexistence results, of a generalized quasilinear Schrödinger potential free equation in the entire 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} involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation has been derived from models of several physical phenomena such as superfluid film in plasma physics as well as the self-channelling of a high-power ultra-short laser in matter. Proof techniques, also in the symmetric setting, are based on variational tools, including concentration compactness principles, to overcome lack of compactness, and the use of a change of variable in order to deal with a well defined functional.

A Critical p(x)-Laplacian Steklov Type Problem with Weights

ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.

Small perturbations of critical nonlocal equations with variable exponents

Abstract In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.

Concentration-compactness type principle for systems with critical terms in [formula omitted]

In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, for nonlinear systems in the context of fractional Sobolev spaces with variable exponents. As an application of the results, we show the existence and asymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with variable exponents and critical growth conditions in RN.

Existence and stability of traveling waves for semi-relativistic Schrödinger equations with van der Waals-type potentials

In this paper, we prove the existence and stability of traveling waves for semi-relativistic Schrödinger equations with van der Waals-type potentials. Using the concentration-compactness principle, we study the corresponding constraint minimization problem of equations and obtain the existence of traveling waves with subcritical arbitrarily small mass. Moreover, we show that the set of boosted ground states is a stable set. Our results contribute to the study of traveling wave solutions and the dynamics of semi-relativistic Schrödinger equations with van der Waals-type potentials.

Almost sharp weighted Sobolev trace inequalities in the unit ball under constraints

In this paper, we establish some improved weighted Sobolev trace inequalities [Formula: see text] under the zero higher order moments constraint via the concentration compactness principle, where [Formula: see text] is a defining function of [Formula: see text] and [Formula: see text]. This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when [Formula: see text].

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.