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Abstract In this paper, we deal with the following fractional p & q $p\&q$ -Laplacian problem: { ( − Δ ) p s u + ( − Δ ) q s u = λ a ( x ) | u | θ − 2 u + μ b ( x ) | u | r − 2 u in Ω , u ( x ) = 0 in R N ∖ Ω , $$ \left \{ \textstyle\begin{array}{l@{\quad }l} (-\Delta )_{p}^{s}u +(-\Delta )_{q}^{s}u =\lambda a(x)|u|^{\theta -2}u+ \mu b(x)|u|^{r-2}u&\text{in}\;\ \Omega , \\ u(x)=0 &\text{in}\;\ \mathbb{R}^{N}\setminus \Omega , \end{array}\displaystyle \right . $$ where Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary, s ∈ ( 0 , 1 ) $s\in (0,1)$ , ( − Δ ) m s $(-\Delta )_{m}^{s}$ ( m ∈ { p , q } ) $(m\in \{p,q\})$ is the fractional m-Laplacian operator, p , q , r , θ ∈ ( 1 , p s ∗ ] $p,q,r,\theta \in (1,p_{s}^{*}]$ , p s ∗ = N p N − s p $p_{s}^{*}=\frac{Np}{N-sp}$ , λ , μ > 0 $\lambda , \mu >0$ , and the weights a ( x ) $a(x)$ and b ( x ) $b(x)$ 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 = p s ∗ $r=p_{s}^{*}$ . Moreover, for the subcritical case r < p s ∗ $r< p_{s}^{*}$ , we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem.

In this paper, we establish some improved weighted Sobolev trace inequalities H 1 (ρ 1 − 2 σ , n + 1) ↪ L q ( n) under the zero higher order moments constraint via the concentration compactness principle, where ρ is a defining function of n + 1 and σ ∈ (0 , 1). 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 σ ∈ (0 , 1 2 ]. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following fractional p & q -Laplacian problem: { (− Δ) p s u + (− Δ) q s u = λ a (x) | u | θ − 2 u + μ b (x) | u | r − 2 u in Ω , u (x) = 0 in R N ∖ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary, s ∈ (0 , 1) , (− Δ) m s (m ∈ { p , q }) is the fractional m-Laplacian operator, p , q , r , θ ∈ (1 , p s ∗ ] , p s ∗ = N p N − s p , λ , μ > 0 , and the weights a (x) and b (x) 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 = p s ∗ . Moreover, for the subcritical case r < p s ∗ , we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

In this paper, we study the following Kirchhoff-type system: $ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $ where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.

In this paper, we establish an improved Hardy–Littlewood–Sobolev inequality on S n under higher-order moments constraint. Moreover, by constructing precise test functions, using improved Hardy–Littlewood–Sobolev inequality on S n , we show such inequality is almost optimal in critical case. As an application, we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality. [ABSTRACT FROM AUTHOR]

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.

In this paper, we study the following Kirchhoff-type system: where and The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle. [ABSTRACT FROM AUTHOR]

In this article, we study the following bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth: $ \begin{equation*} \begin{cases} -\left( \int_{\Omega}|\nabla u|^2dx\right)^r\Delta u+\phi u = u^5+\lambda\left( \int_{\Omega}F(x, u)dx\right)^sf(x, u), & \mathrm{in}\ \ \Omega, \\ -\Delta\phi = u^2, u>0, & \mathrm{in}\ \ \Omega, \\ u = \phi = 0, & \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $ where $ \Omega\subset \mathbb{R}^3 $ is a smooth bounded domain, $ \lambda > 0 $, $ 0\leq r < 1 $, $ 0 < s < \frac{1-r}{3(r+1)} $ and $ f(x, u) $ satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.

This paper intends to study the following degenerate fractional Schrödinger–Kirchhoff–Poisson equations with critical nonlinearity and electromagnetic fields in R 3 : { ε 2 s M ([ u ] s , A 2) (− Δ) A s u + V (x) u + ϕ u = k (x) | u | r − 2 u + (I μ ∗ | u | 2 s ♯ ) | u | 2 s ♯ − 2 u , x ∈ R 3 , (− Δ) t ϕ = u 2 , x ∈ R 3 , where ε > 0 is a positive parameter, 3 / 4 < s < 1 , 0

In this paper, our major concern is the existence of normalized solutions for a Choquard equation with a L 2 -subcritical growth. As a result, we prove that the normalized solutions and ground state solutions are equivalent by constructing Pohozaev sequence and combining the principle of compactness concentration. [ABSTRACT FROM AUTHOR]

Copyright of Basic Sciences Journal of Textile Universities / Fangzhi Gaoxiao Jichu Kexue Xuebao is the property of Basic Sciences Journal of Textile Universities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, we study the existence of infinitely many nontrivial solutions for the following doubly nonlocal problem involving fractional (p (x) , p +) -Laplacian. (− Δ) p (x) s u + (− Δ) p + s u = | u | r (x) − 2 u + | u | p s ∗ (x) − 2 u + λ f (x , u) in Ω , u = 0 in R N ∖ Ω. Here, Ω ⊂ R N is a bounded Lipschitz domain, λ > 0 , q s ∈ (0 , 1) , p (⋅ , ⋅) is a continuous, bounded, symmetric function in R N × R N such that p + = sup (x , y) ∈ R N × R N { p (x , y) } < N s , r ∈ C (Ω ¯) , p s ∗ (x) = N p (x , x) N − s p (x , x) for every x ∈ R N and the function f satisfies certain assumptions which will be made precise later. Further, the exponents p s ∗ (⋅) and r (⋅) are two critical exponents with the assumption that the critical sets { x ∈ Ω : p s ∗ (x) = (p +) s ∗ } and { x ∈ Ω : r (x) = (p +) s ∗ } are nonempty, where (p +) s ∗ = N p + N − s p + . We also develop a concentration compactness type principle in the process. [ABSTRACT FROM AUTHOR]

We analyze the existence and non-existence of cylindrical solutions as well as ground state solutions to a class of Kirchhoff-type problems with critical Sobolev exponent, subcritical Sobolev exponent, and critical Sobolev–Hardy exponent. To the best of our knowledge, this is the first time to study the existence of cylindrically symmetric solutions to the Kirchhoff equations in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of critical p(x)-Kirchhoff problems with a singular term. A nontrivial positive solution is obtained by combining variational methods with an appropriate truncation argument. [ABSTRACT FROM AUTHOR]

We give a new proof of the almost sharp Moser–Trudinger inequality on smooth compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces and generalize it to Riemannian manifolds with continuous metrics and to higher order Sobolev spaces on manifolds with boundary under several boundary conditions. These generalizations can be applied to fourth order Q curvature equations in dimension 4. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of stable standing waves for the nonlinear Schr\"{o}dinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities \[ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi=0,~~(t,x)\in [0,T^\star)\times \mathbb{R}^N. \] By using concentration compactness principle, when one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper \cite {13} to the $L^2$-critical and $L^2$-supercritical nonlinearities.

In this paper, we study the nonlinear fractional Schrödinger–Poisson system (−Δ)su+V(x)u+ϕu=f(u)inℝ3,(−Δ)sϕ=u2inℝ3,where (− Δ)s is the fractional Laplacian of order s, 34

In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:

Abstract In 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.

This article concerns the existence of positive solutions for the second order equation involving a nonlocal term -Δu = γ(-Δ)-1u + |u|p-1u, under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter γ>0, and up to the critical value of the exponent p, i.e. when 1

In the framework of fractional Sobolev space, using Nehari manifold and concentration compactness principle, we study a minimization problem in the whole space involving the fractional Laplacian. Firstly, we give a Lions type lemma in fractional Sobolev space, which is crucial in the proof of our main result. Then, by showing a relative compactness of minimizing sequence, we obtain the existence of minimizer for the above-mentioned fractional minimization problem. Furthermore, we also point out that the minimizer is actually a ground state solution for the associated fractional Schrodinger equation

By using the mountain pass lemma and a concentration compactness principle, we obtain the existence of positive solutions to the fractional elliptic problem with two critical Hardy-Sobolev exponents at the origin.

In this article, we use variational methods to prove the existence of positive solutions for an elliptic equation with two critical Hardy Sobolev exponents at the same pole and a certain nonlinear perturbation. Some parameters play a crucial role in our work. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-cooperative elliptic systems of Kirchhoff type involving p -biharmonic operator and critical growth. With the help of the Limit index theory due to Li (Nonlinear Anal. TMA 30(7):4619–4627, 1997) and the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, kinds of Schrödinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo–Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in L 2 -subcritical and L 2 -critical cases are established in a systematic way. [ABSTRACT FROM AUTHOR]

In this article, we explore the existence of multiple solutions for a p-Kirchhoff equation with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle and Ekeland's variational principle, we obtain two positive weak solutions.

This paper is devoted to study the existence of multiple solutions for the following fractional p -Kirchhoff problem (0.1) M ∫ R 2 n | u (x) − u (y) | p | x − y | n + p s d x d y (− △) p s u = λ | u | q − 2 u + | u | r − 2 u | x | α , in Ω , u = 0 , in R n ∖ Ω , where (− △) p s denotes the fractional p -Laplace operator, Ω is a smooth bounded set in R n containing 0 with Lipschitz boundary, M (t) = a + b t θ − 1 with a ≥ 0 , b > 0 , θ > 1. λ > 0 , 1 < q < p < θ p ≤ r ≤ p α ∗ , p α ∗ = (n − α) p n − p s is the fractional critical Hardy–Sobolev exponent for 0 ≤ α < p s < n. By using fibering maps and Nehari manifold, we obtain that the existence of multiple solutions to problem (0.1) for both Hardy–Sobolev subcritical and critical cases. In particular, the concentration compactness principle will be used to overcome the lack of compactness for the critical case. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Henan University of Science & Technology, Natural Science is the property of Editorial Office of Journal of Henan University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we establish the existence of extremals for two kinds of Stein–Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein–Weiss inequalities with full weights in Theorem 1.1 and the Stein–Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein–Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein–Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein–Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5). [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonhomogeneous p-Laplacian el- liptic equations with a critical Sobolev exponent and multiple Hardy type terms. By the Ekeland variational principale on a Nehari manifold and the mountain pass lemma, we prove the existence of multiple solutions, under sufficient condi- tions on the data and the considered parameters. [ABSTRACT FROM AUTHOR]

In this work, we study the following critical problem involving the fractional Laplacian: (−Δ)su−μu|x|2s=λg(x)up+K(x)u2s∗−1,x∈RN,where s  ∈  (0,1), N  >  2s, 0

Abstract In this paper we prove the existence of a nontrivial τ -antisymmetric solution for the following system { − Δ u + u = | u | 2 p − 2 u + β (x) | v | p | u | p − 2 u , in R N − Δ v + ω 2 v = | v | 2 p − 2 v + β (x) | u | p | v | p − 2 v , in R N where N ≥ 3 , 2 < 2 p < 2 ⁎ , ω > 0 is a positive constant and τ : R N → R N is a nontrivial orthogonal involution. Here β is a continuous function, invariant under τ , satisfying additional conditions, some of them related to ω. The basic tool employed here is the Concentration Compactness Principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equations: [ABSTRACT FROM AUTHOR]

We study the problem of solving some general integral formulas and then apply the conclusions to obtain results about the existence of weak solutions of various degenerate p -Laplacian equations. We adapt Variational Calculus methods and the Mountain Pass Lemma without the Palais–Smale condition, and we use an abstract version of Lions' Concentration Compactness Principle II. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent ... where 1 < q < 2 and λ > 0. Under some appropriate conditions on l and h, we show that there exists λ > 0 such that the above problem has at least two positive solutions for each λ ∈ (0, λ*) by using the Mountain Pass Theorem and Ekeland's Variational Principle. [ABSTRACT FROM AUTHOR]

Consider the following fractional Kirchhoff equations involving critical exponent: [ABSTRACT FROM AUTHOR]

The paper deals with the following equation of Kirchhoff type, - (1 + b(∫R3∣▿ u∣u2dx)r) ▿u + u = k(x) -∣u∣q-2u +θ(g(u)+ λh(x)u with x ∊ R3, where u ∊ H1(R3), b > 0; 0 < r < 2; q ∊ [2(r + 1), 6),θ is a small constant, λ is a parameter, and a weight function h(x) ≥ 0. It is known that the linear operator -Δu + u - λh(x)u is coercive if 0 < λ < λ1(h) and is non-coercive if λ > λ1(h), where λ1(h) is the first eigenvalue of the operator -Δu + u with the weight h(x). Under suitable conditions on the functions k(x) and g(s), it is shown that the equation has a positive solution for any λ ∊ (0; λ1(h)) and two positive solutions for λ ∊ (λ1(h); λ1(h) + ~δ) with ~δ > 0 small. The conditions imposed on k(x) and g(s) are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity results of positive solutions for a Kirchhoff type problem with critical growth, with the help of the concentration compactness principle, and we prove that problem admits two positive solutions, and one of the solutions is a positive ground state solution. [ABSTRACT FROM AUTHOR]

In this paper, based on the theory of variable exponent space, we study a class of minimizing problem on Nehari manifold via concentration compactness principle. Under suitable assumptions, by showing a relative compactness of minimizing sequences, we prove the existence of minimizers. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: ( − Δ ) α u + V ( x ) u = k ( x ) f ( u ) + λ | u | 2 α ∗ − 2 u in R N , where ( − Δ ) α is the fractional Laplacian operator with α ∈ ( 0 , 1 ) , N ≥ 2 , λ is a positive real parameter and 2 α ∗ = 2 N / ( N − 2 α ) is the critical Sobolev exponent, V ( x ) and k ( x ) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, with the aid of variational method and concentration-compactness principle, a positive ground state solution is obtained for a class of Kirchhoff type equations with critical growth { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u = u 5 + λ k ( x ) u q − 1 , x ∈ R 3 , u ∈ D 1 , 2 ( R 3 ) , where λ > 0 is a parameter, 2 < q < 6 and k satisfies some conditions. [ABSTRACT FROM AUTHOR]