Concentration-compactness Principle Research Articles

On the noncooperative Schrödinger–Kirchhoff system involving the critical nonlinearities on the Heisenberg group

This paper deals with the existence of solutions for the noncooperative Schrödinger–Kirchhoff system involving the p-Laplacian operator and critical nonlinearities on the Heisenberg group. Under some suitable conditions, together with the limit index theory and the concentration–compactness principle, we obtain the existence and multiplicity of solutions for this system. To our best knowledge, the existence results for the noncooperative system with p-Laplacian and critical nonlinearities are new on the Heisenberg group.

Critical Kirchhoff equations involving the p-sub-Laplacians operators on the Heisenberg group

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the [Formula: see text]-sub-Laplacians operators on the Heisenberg group of the form [Formula: see text] where [Formula: see text] is the [Formula: see text]-sub-Laplacian, [Formula: see text], [Formula: see text] is the horizontal Sobolev space on [Formula: see text]. And [Formula: see text] is the homogeneous dimension of [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function [Formula: see text], the potential function [Formula: see text] and [Formula: see text], together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the [Formula: see text]-Laplacian type operators on the Heisenberg group.

Ground state solution for a nonlinear fractional magnetic Schrödinger equation with indefinite potential

This paper is concerned with the following nonlinear fractional Schrödinger equation with a magnetic field: ε2s(−Δ)A/εsu+V(x)u=f(|u|2)u inRN, where ɛ > 0 is a parameter, s ∈ (0, 1), N ≥ 3, V:RN→R and A:RN→RN are continuous potentials, and V may be sign-changing; the nonlinearity is superlinear with subcritical growth but without satisfying the Ambrosetti–Rabinowitz condition. Based on the Nehari manifold method, concentration-compactness principle, and variational methods, we prove the existence of a ground state solution for the above equation when ɛ is sufficiently small. Our results improve and extend the result of Ambrosio and d’Avenia [J. Differ. Equations 264, 3336–3368 (2018)].

Multiplicity of Solutions for Fractional-Order Differential Equations via the κ(x)-Laplacian Operator and the Genus Theory

In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.

Multiple positive solutions for Kirchhoff-type equations involving critical-concave nonlinearities

In this paper, we research the following Kirchhoff type problem with critical growth (1) 0, & u\\in H^1(\\mathbb{R}^{3}), \\end{cases} \\end{equation}$$]]> { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + u = λ f ( x ) | u | q − 2 u + | u | 4 u , x ∈ R 3 , u > 0 , u ∈ H 1 ( R 3 ) , where 1<q<2, a>0 is a positive constant, 0 $ ]]> b , λ > 0 are real parameters and f ∈ L 6 6 − q ( R 3 ) is a nonzero nonnegative function. We prove the problem has at least two positive solutions by using concentration compactness principle and variational method when b is sufficiently small. Moreover, we also explore the asymptotic behaviour of the positive solutions as b n → n 0 + .

Existence and multiplicity of solutions for critical nonlocal equations with variable exponents

ABSTRACT In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: where M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that , where is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases.

Degenerate Fractional Kirchhoff-Type System with Magnetic Fields and Upper Critical Growth

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak {M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak {M}(\Vert v\Vert _{s,A})[(-\Delta )^s_Av+v] = G_v(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|v|^{p^*}\right) |v|^{p^*-2}v \ &{}\text{ in }\,\,\mathbb {R}^N, \end{array}\right. \end{aligned}$$where $$\begin{aligned}\Vert u\Vert _{s,A}=\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}{\text {d}}x {\text {d}}y+\int _{\mathbb {R}^N}|u|^2{\text {d}}x\right) ^{1/2},\end{aligned}$$and \((-\Delta )_{A}^s\) and A are called magnetic operator and magnetic potential, respectively, \(\mathfrak {M}:\mathbb {R}^{+}_{0}\rightarrow \mathbb {R}^{+}_0\) is a continuous Kirchhoff function, \(\mathcal {I}_\mu (x) = |x|^{N-\mu }\) with \(0<\mu <N\), \(C^1\)-function G satisfies some suitable conditions, and \(p^* =\frac{N+\mu }{N-2s}\). We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.

Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system

In this paper, we analyze the stability and instability of standing waves for a generalized Zakharov-Rubenchik system (or the Benney-Roskes system) in spatial dimensions N = 2, 3. We show that the standing waves generated by the set of minimizers for the associated variational problem are stable, for N = 2 and σ(p − 2) &gt; 0. We also show that the standing waves are strongly unstable, for N = 3 and if either σ &lt; 0 and 4/3 &lt;p&lt; 4, or σ &gt; 0 and 0 &lt;p&lt; 2. Results follow by using the variational characterization of standing waves, the concentration compactness principle due to J. Lions and the compactness lemma due to E. Lieb to solve the associated minimization problem.

The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L 2-subcritical and L 2-supercritical cases

Abstract This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u &gt; 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where 0 &lt; s &lt; 1 0\lt s\lt 1 , a a , μ &gt; 0 \mu \gt 0 , N ≥ 2 N\ge 2 , and 2 &lt; p &lt; 2 s ∗ 2\lt p\lt {2}_{s}^{\ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) \left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) \left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

Cylindrically symmetric solutions of curl–curl equation with nonlocal nonlinearity

In this paper we consider a curl–curl equation with nonlocal nonlinearity involving Riesz potential. Under the assumption of 0∉σ(L), our investigation becomes a strong indefinite problem. Based on the method of constraining Nehari–Pankov manifold and concentration compactness principle, we get the cylindrically symmetric solution to curl–curl equation.

Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions

In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.

Ground states for Schrödinger–Kirchhoff equations of fractional [formula omitted]-Laplacian involving logarithmic and critical nonlinearity

For 0<s<1<p<N/s and a parameter λ>0, we consider a class of Schrödinger–Kirchhoff equations of fractional p-Laplacian involving logarithmic nonlinearity and critical exponential growth: mup−Δpsu+Vxup−2u=λhxuθp−2ulnu+σups∗−2uinRNwith u=∫R2Nux−uypx−yN+spdxdy+∫RNVxupdx1/p,where ps∗=Np/N−sp, 1≤θ<ps∗/p, σ∈0,1, m(⋅) is a Kirchhoff function, V is a Rabinowitz potential, and h is a positive bounded perturbation. By using concentration-compactness principle and minimax argument, we prove the existence of a non-trivial ground state solution for λ sufficiently small.

Existence of Solutions for p-Kirchhoff Problem of Brézis-Nirenberg Type with Singular Terms

In this paper, we prove the existence of positive solution for a p-Kirchhoff problem of Brézis-Nirenberg type with singular terms, nonlocal term, and the Caffarelli-Kohn-Nirenberg exponent by using variational methods, concentration compactness, and maximum principle.

On fractional logarithmic Schrödinger equations

Abstract We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 &lt; s &lt; 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N}) . Under different assumptions on the potential V ( x ) V\left(x) , we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) , and inspired by Cazenave (Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127–1140), we first prove that the variational functional is well defined in a subspace of H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) . Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results.

On the p-Laplacian Kirchhoff–Schrödinger equation with potentials vanishing or unbounded at infinity in R3

This paper mainly deals with a class of the p-Laplacian Kirchhoff–Schrödinger equation with potentials vanishing or unbounded at infinity. We prove the existence and concentration behavior of positive solutions for the problem by the variational methods and concentration-compactness principle. The main feature of this paper is that the potential decays to zero at infinity like |x|−α with 0 &amp;lt; α ≤ 2, and the function K(x) is allowed to be unbounded, which is different from some previous papers.

On degenerate fractional Schrödinger–Kirchhoff–Poisson equations with upper critical nonlinearity and electromagnetic fields

This paper intends to study the following degenerate fractional Schrödinger–Kirchhoff–Poisson equations with critical nonlinearity and electromagnetic fields in : where is a positive parameter, , 0<t<1, V is an electric potential satisfying suitable assumptions, and , with , and . With the help of the concentration compactness principle and variational method, and together with some careful analytical skills, we prove the existence and multiplicity of solutions for the above problem as in degenerate cases, that is the Kirchhoff term M can vanish at zero.

On elliptic equations with Stein–Weiss type convolution parts

The aim of this paper is to study the critical elliptic equations with Stein–Weiss type convolution parts $$\begin{aligned} \displaystyle -\Delta u =\frac{1}{|x|^{\alpha }}\left( \int _{\mathbb {R}^{N}}\frac{|u(y)|^{2_{\alpha , \mu }^{*}}}{|x-y|^{\mu }|y|^{\alpha }}dy\right) |u|^{2_{\alpha , \mu }^{*}-2}u,\quad x\in \mathbb {R}^{N}, \end{aligned}$$where the critical exponent is due to the weighted Hardy–Littlewood–Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, the subcritical case is also considered, the existence, symmetry, regularity of the positive solutions are obtained.

Existence of solutions for critical (p,q)-Laplacian equations in ℝN

In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([Formula: see text])-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter [Formula: see text] are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter [Formula: see text] is far from [Formula: see text]. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.

POSITIVE SOLUTIONS FOR A NONLOCAL PROBLEM WITH CRITICAL SOBOLEV EXPONENT IN HIGHER DIMENSIONS

This paper is devoted to a nonlocal problem involving critical Sobolev exponent and negative nonlocal term. By virtue of a cut-off technique and the concentration compactness principle, we prove the existence and asymptotic behavior of positive solutions for the considered problem. In particular, our results generalize the existence results of positive solutions to higher dimensions <i>N</i> ≥ 5.

Positive ground state of coupled planar systems of nonlinear Schrödinger equations with critical exponential growth

In this paper, we prove the existence of a positive ground state solution to the following coupled system involving nonlinear Schrödinger equations: { − Δ u + V 1 ( x ) u = f 1 ( x , u ) + λ ( x ) v , x ∈ R 2 , − Δ v + V 2 ( x ) v = f 2 ( x , v ) + λ ( x ) u , x ∈ R 2 , where λ , V 1 , V 2 ∈ C ( R 2 , ( 0 , + ∞ ) ) and V 1 ( x ) have critical exponential growth in the sense of Trudinger–Moser inequality. The potentials V 1 ( x ) and V 2 ( x ) satisfy a condition involving the coupling term λ ( x ) , namely 0 &lt; λ ( x ) ≤ λ 0 V 1 ( x ) V 2 ( x ) . We use non-Nehari manifold, Lions's concentration compactness and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap regularity lifting argument and L q -estimates we get regularity and asymptotic behavior. Our results improve and extend the previous results.