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Journal articles on the topic 'Ekeland''s Variational Principle'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

KANDILAKIS, D. A., M. MAGIROPOULOS, and N. ZOGRAPHOPOULOS. "EXISTENCE AND BIFURCATION RESULTS FOR FOURTH-ORDER ELLIPTIC EQUATIONS INVOLVING TWO CRITICAL SOBOLEV EXPONENTS." Glasgow Mathematical Journal 51, no. 1 (2009): 127–41. http://dx.doi.org/10.1017/s0017089508004588.

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AbstractLet Ω be a smooth bounded domain in RN, with N ≥ 5. We provide existence and bifurcation results for the elliptic fourth-order equation Δ2u − Δpu = f(λ, x, u) in Ω, under the Dirichlet boundary conditions u = 0 and ∇u = 0. Here λ is a positive real number, 1 < p ≤ 2# and f(.,., u) has a subcritical or a critical growth s, 1 < s ≤ 2*, where $2^{\ast}:=\frac{2N}{N-4}$ and $2^{\#}:=\frac{2N}{N-2}$. Our approach is variational, and it is based on the mountain-pass theorem, the Ekeland variational principle and the concentration-compactness principle.

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2

Aghayeva, Charkaz. "Stochastic optimal control problem of constrained switching system with delay." Filomat 30, no. 3 (2016): 711–20. http://dx.doi.org/10.2298/fil1603711a.

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This paper concerns the stochastic optimal control problem of switching systems with delay. The evolution of the system is governed by the collection of stochastic delay differential equations with initial conditions that depend on its previous state. The restriction on the system is defined by the functional constraint that contains state and time parameters. First, maximum principle for stochastic control problem of delay switching system without constraint is established. Finally, using Ekeland?s variational principle, the necessary condition of optimality for control system with constraint

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3

BENAHMED, S., and D. AZÉ. "ON FIXED POINTS OF GENERALIZED SET-VALUED CONTRACTIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (2009): 16–22. http://dx.doi.org/10.1017/s000497270900046x.

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AbstractUsing a variational method introduced in [D. Azé and J.-N. Corvellec, ‘A variational method in fixed point results with inwardness conditions’, Proc. Amer. Math. Soc.134(12) (2006), 3577–3583], deriving directly from the Ekeland principle, we give a general result on the existence of a fixed point for a very general class of multifunctions, generalizing the recent results of [Y. Feng and S. Liu, ‘Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings’, J. Math. Anal. Appl.317(1) (2006), 103–112; D. Klim and D. Wardowski, ‘Fixed point theorems

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4

Pucci, Patrizia, Mingqi Xiang, and Binlin Zhang. "Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian." Advances in Calculus of Variations 12, no. 3 (2019): 253–75. http://dx.doi.org/10.1515/acv-2016-0049.

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AbstractThe paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case(a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N},where\|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}},{a,b\in\mathbb{R}^{+}_{0}}, with

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5

WU, Bo. "Note on Ekeland Variational Principle." Acta Analysis Functionalis Applicata 12, no. 3 (2011): 216–20. http://dx.doi.org/10.3724/sp.j.1160.2010.00216.

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6

Suzuki, Tomonari. "The strong Ekeland variational principle." Journal of Mathematical Analysis and Applications 320, no. 2 (2006): 787–94. http://dx.doi.org/10.1016/j.jmaa.2005.08.004.

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7

Alfuraidan, Monther Rashed, and Mohamed Amine Khamsi. "Ekeland variational principle on weighted graphs." Proceedings of the American Mathematical Society 147, no. 12 (2019): 5313–21. http://dx.doi.org/10.1090/proc/14642.

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8

Mingqi, Xiang, Vicenţiu D. Rădulescu, and Binlin Zhang. "Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 3 (2018): 1249–73. http://dx.doi.org/10.1051/cocv/2017036.

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In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system: [see formula in PDF] where (–Δ)sp is the fractional p–Laplace operator with 0 < s < 1 < p < N/s, α, β > 1 with α + β =p*s, M : ℝ+0 → ℝ+0 is a continuous function, V : ℝN → ℝ+ is a continuous function, λ > 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for t

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9

Bu, Weichun, Tianqing An, Guoju Ye, and Said Taarabti. "Negative Energy Solutions for a New Fractional p x -Kirchhoff Problem without the (AR) Condition." Journal of Function Spaces 2021 (March 22, 2021): 1–13. http://dx.doi.org/10.1155/2021/8888078.

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In this paper, we investigate the following Kirchhoff type problem involving the fractional p x -Laplacian operator. a − b ∫ Ω × Ω u x − u y p x , y / p x , y x − y N + s p x , y d x d y L u = λ u q x − 2 u + f x , u x ∈ Ω u = 0 x ∈ ∂ Ω , , where Ω is a bounded domain in ℝ N with Lipschitz boundary, a ≥ b > 0 are constants, p x , y is a function defined on Ω ¯ × Ω ¯ , s ∈ 0 , 1 , and q x > 1 , L u is the fractional p x -Laplacian operator, N > s p x , y , for any x , y ∈ Ω ¯ × Ω ¯ , p x ∗ = p x , x N / N − s p x , x , λ is a given positive parameter, and f is a continuous function. By

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10

Kim, Yun-Ho. "Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN". Mathematics 8, № 10 (2020): 1792. http://dx.doi.org/10.3390/math8101792.

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We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a resu

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11

Qu, De-ning, and Cao-zong Cheng. "Ekeland Variational Principle for Generalized Vector Equilibrium Problems with Equivalences and Applications." Journal of Function Spaces and Applications 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/267585.

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The aim of this paper is to introduce Ekeland variational principle with variants for generalized vector equilibrium problems and to establish some existence results of solutions of generalized vector equilibrium problems with compact or noncompact domain as applications. Finally, some equivalent results of the established Ekeland variational principle are presented.

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12

Cuesta, Mabel. "Minimax theorems onC1manifolds via Ekeland variational principle." Abstract and Applied Analysis 2003, no. 13 (2003): 757–68. http://dx.doi.org/10.1155/s1085337503303100.

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We prove two minimax principles to find almost critical points ofC1functionals restricted to globally definedC1manifolds of codimension1. The proof of the theorems relies on Ekeland variational principle.

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13

Farkas, Csaba. "A generalized form of Ekeland’s variational principle." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 1 (2012): 101–12. http://dx.doi.org/10.2478/v10309-012-0008-5.

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Abstract In this paper we prove a generalized version of the Ekeland variational principle, which is a common generalization of Zhong variational principle and Borwein Preiss Variational principle. Therefore in a particular case, from this variational principle we get a Zhong type variational principle, and a Borwein-Preiss variational principle. As a consequence, we obtain a Caristi type fixed point theorem.

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14

Xiang, Mingqi, Binlin Zhang, and Massimiliano Ferrara. "Multiplicity results for the non-homogeneous fractional p -Kirchhoff equations with concave–convex nonlinearities." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2177 (2015): 20150034. http://dx.doi.org/10.1098/rspa.2015.0034.

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In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p -Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities: a + b ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p d x d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in R N , where ( − Δ ) p s is the fractional p -Laplace operator, a + b >0 with a , b ∈ R 0 + , λ>0 is a real parameter, 0 < s < 1 < p < ∞ with sp

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15

Qiu, Jing-Hui. "A pre-order principle and set-valued Ekeland variational principle." Journal of Mathematical Analysis and Applications 419, no. 2 (2014): 904–37. http://dx.doi.org/10.1016/j.jmaa.2014.05.027.

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16

Beer, Gerald, and Asen L. Dontchev. "The weak Ekeland variational principle and fixed points." Nonlinear Analysis: Theory, Methods & Applications 102 (June 2014): 91–96. http://dx.doi.org/10.1016/j.na.2014.01.022.

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17

Cobzaş, S. "Ekeland Variational Principle in asymmetric locally convex spaces." Topology and its Applications 159, no. 10-11 (2012): 2558–69. http://dx.doi.org/10.1016/j.topol.2012.04.015.

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18

Lin, Lai-Jiu, Chih-Sheng Chuang, and Suliman Al-Homidan. "Ekeland type variational principle with applications to quasi-variational inclusion problems." Nonlinear Analysis: Theory, Methods & Applications 72, no. 2 (2010): 651–61. http://dx.doi.org/10.1016/j.na.2009.07.038.

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19

Bonanno, Gabriele. "A critical point theorem via the Ekeland variational principle." Nonlinear Analysis: Theory, Methods & Applications 75, no. 5 (2012): 2992–3007. http://dx.doi.org/10.1016/j.na.2011.12.003.

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20

Cobzaş, S. "Completeness in quasi-metric spaces and Ekeland Variational Principle." Topology and its Applications 158, no. 8 (2011): 1073–84. http://dx.doi.org/10.1016/j.topol.2011.03.003.

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21

Qiu, Jinghui. "Vectorial Ekeland Variational Principle and Cyclically Antimonotone Equilibrium Problems." Acta Mathematica Scientia 39, no. 2 (2019): 524–44. http://dx.doi.org/10.1007/s10473-019-0216-4.

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22

Attouch, H., and G. Beer. "Stability of the geometric Ekeland variational principle: Convex case." Journal of Optimization Theory and Applications 81, no. 1 (1994): 1–19. http://dx.doi.org/10.1007/bf02190310.

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23

Maia, Liliane A., Olimpio H. Miyagaki, and Sergio H. M. Soares. "A Sign-Changing Solution for an Asymptotically Linear Schrödinger Equation." Proceedings of the Edinburgh Mathematical Society 58, no. 3 (2015): 697–716. http://dx.doi.org/10.1017/s0013091514000339.

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AbstractThe aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

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24

Yucedag, Zehra, Mustafa Avci, and Rabil Mashiyev. "ON AN ELLIPTIC SYSTEM OF P(X)-KIRCHHOFF-TYPE UNDER NEUMANN BOUNDARY CONDITION." Mathematical Modelling and Analysis 17, no. 2 (2012): 161–70. http://dx.doi.org/10.3846/13926292.2012.655788.

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In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution.

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25

Alfuraidan, Monther R., та Mohamed A. Khamsi. "Ekeland Variational Principle in the Variable Exponent Sequence Spaces ℓp(·)". Mathematics 8, № 3 (2020): 375. http://dx.doi.org/10.3390/math8030375.

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In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces ℓ p ( · ) . The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the establishment of the classical EVP, that has hitherto prevented a successful treatment of the modular case. As an application, we establish a modular version of Caristi’s fixed point theorem in ℓ p ( · ) .

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26

Alleche, Boualem, and Vicenţiu D. Rădulescu. "The Ekeland variational principle for equilibrium problems revisited and applications." Nonlinear Analysis: Real World Applications 23 (June 2015): 17–25. http://dx.doi.org/10.1016/j.nonrwa.2014.11.006.

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27

Ha, Truong Xuan Duc. "The Ekeland variational principle for set-valued maps involving coderivatives." Journal of Mathematical Analysis and Applications 286, no. 2 (2003): 509–23. http://dx.doi.org/10.1016/s0022-247x(03)00482-7.

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28

Zhang, Chuang-liang, and Nan-jing Huang. "Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems." Optimization 69, no. 6 (2019): 1255–80. http://dx.doi.org/10.1080/02331934.2019.1689978.

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29

Lin, Lai-Jiu, Sung-Yu Wang, and QamrulHasan Ansari. "Critical Point Theorems and Ekeland Type Variational Principle with Applications." Fixed Point Theory and Applications 2011, no. 1 (2011): 914624. http://dx.doi.org/10.1155/2011/914624.

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30

Zhu, Jiang, Lei Wei, Yeol Je Cho, and Cheng Cheng Zhu. "Vectorial Ekeland Variational Principles and Inclusion Problems in Cone Quasi-Uniform Spaces." Abstract and Applied Analysis 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/310369.

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Some new vectorial Ekeland variational principles in cone quasi-uniform spaces are proved. Some new equivalent principles, vectorial quasivariational inclusion principle, vectorial quasi-optimization principle, vectorial quasiequilibrium principle are obtained. Also, several other important principles in nonlinear analysis are extended to cone quasi-uniform spaces. The results of this paper extend, generalize, and improve the corresponding results for Ekeland's variational principles of the directed vectorial perturbation type and other generalizations of Ekeland's variational principles in th

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31

CORRÊA, FRANCISCO JULIO S. A., and AUGUSTO CÉSAR DOS REIS COSTA. "A VARIATIONAL APPROACH FOR A BI-NON-LOCAL ELLIPTIC PROBLEM INVOLVING THE p(x)-LAPLACIAN AND NON-LINEARITY WITH NON-STANDARD GROWTH." Glasgow Mathematical Journal 56, no. 2 (2013): 317–33. http://dx.doi.org/10.1017/s001708951300027x.

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AbstractIn this paper we are concerned with a class of p(x)-Kirchhoff equation where the non-linearity has non-standard growth and contains a bi-non-local term. We prove, by using variational methods (Mountain Pass Theorem and Ekeland Variational Principle), several results on the existence of positive solutions.

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32

Hassani, Abdessamad, and Khalid Iskafi. "Schrödinger–Poisson system with potential of critical growth." Asian-European Journal of Mathematics 09, no. 04 (2016): 1650086. http://dx.doi.org/10.1142/s1793557116500868.

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In this paper, we consider a Schrödinger–Poisson system with nonlinear potential of critical growth, and we prove existence of positive solution with positive energy by using the Ekeland variational principle and the Mountain-Pass theorem.

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33

Qiu, Jing-Hui. "A generalized Ekeland vector variational principle and its applications in optimization." Nonlinear Analysis: Theory, Methods & Applications 71, no. 10 (2009): 4705–17. http://dx.doi.org/10.1016/j.na.2009.03.034.

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34

Suzuki, Tomonari. "Characterizations of reflexivity and compactness via the strong Ekeland variational principle." Nonlinear Analysis: Theory, Methods & Applications 72, no. 5 (2010): 2204–9. http://dx.doi.org/10.1016/j.na.2009.10.019.

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35

Georgiev, Pando Grigorov. "The strong Ekeland variational principle, the strong Drop theorem and applications." Journal of Mathematical Analysis and Applications 131, no. 1 (1988): 1–21. http://dx.doi.org/10.1016/0022-247x(88)90187-4.

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36

Ha, Truong Xuan Duc. "The Ekeland variational principle for Henig proper minimizers and super minimizers." Journal of Mathematical Analysis and Applications 364, no. 1 (2010): 156–70. http://dx.doi.org/10.1016/j.jmaa.2009.10.065.

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37

Yang, Liu. "Multiplicity of Homoclinic Solutions for a Class of Nonperiodic Fourth-Order Differential Equations with General Perturbation." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/126435.

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In this paper, we investigate a class of nonperiodic fourth-order differential equations with general perturbation. By using the mountain pass theorem and the Ekeland variational principle, we obtain that such equations possess two homoclinic solutions. Recent results in the literature are generalized and significantly improved.

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38

Alves, Claudianor O., Marcos L. M. Carvalho, and José V. A. Gonçalves. "On existence of solution of variational multivalued elliptic equations with critical growth via the Ekeland principle." Communications in Contemporary Mathematics 17, no. 06 (2015): 1450038. http://dx.doi.org/10.1142/s0219199714500382.

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We study the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sob

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39

Tavani, M. R. H. "Existence of three weak solutions for fourth-order elastic beam equations on the whole space." Ukrains’kyi Matematychnyi Zhurnal 72, no. 12 (2020): 1697–707. http://dx.doi.org/10.37863/umzh.v72i12.881.

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UDC 517.9 Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, <em>A critical point theorem via the Ekeland variational principle</em>, Nonlinear Anal., <strong>75</strong>, 2992-3007 (2012)].

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40

Wu, Yong, Zhenhua Qiao, Mohamed Karim Hamdani, Bingyu Kou, and Libo Yang. "A Class of Variable-Order Fractional p · -Kirchhoff-Type Systems." Journal of Function Spaces 2021 (February 27, 2021): 1–6. http://dx.doi.org/10.1155/2021/5558074.

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This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional p x -operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

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41

Iqbal, Iram, and Nawab Hussain. "Ekeland-type variational principle with applications to nonconvex minimization and equilibrium problems." Nonlinear Analysis: Modelling and Control 24, no. 3 (2019): 407–32. http://dx.doi.org/10.15388/na.2019.3.6.

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The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

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42

Ha, Truong Xuan Duc. "A remark on the lower semicontinuity assumption in the Ekeland variational principle." Optimization 65, no. 10 (2016): 1781–89. http://dx.doi.org/10.1080/02331934.2016.1195830.

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43

Al-Homidan, Suliman, Qamrul Hasan Ansari, and Gabor Kassay. "Vectorial form of Ekeland variational principle with applications to vector equilibrium problems." Optimization 69, no. 3 (2019): 415–36. http://dx.doi.org/10.1080/02331934.2019.1589469.

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44

Ha, T. X. D. "Some Variants of the Ekeland Variational Principle for a Set-Valued Map." Journal of Optimization Theory and Applications 124, no. 1 (2005): 187–206. http://dx.doi.org/10.1007/s10957-004-6472-y.

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45

Qiu, JingHui. "Vectorial Ekeland variational principle for systems of equilibrium problems and its applications." Science China Mathematics 60, no. 7 (2017): 1259–80. http://dx.doi.org/10.1007/s11425-015-9005-4.

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46

Barbu, Viorel. "A variational approach to nonlinear stochastic differential equations with linear multiplicative noise." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 71. http://dx.doi.org/10.1051/cocv/2018065.

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One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations wit

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47

Meghea, Irina. "Variational Approaches to Characterize Weak Solutions for Some Problems of Mathematical Physics Equations." Abstract and Applied Analysis 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/2071926.

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This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving thep-Laplacian and thep-pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned opera

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48

Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 5 (2001): 1091–111. http://dx.doi.org/10.1017/s0308210500001281.

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In this paper we consider quasilinear hemivariational inequalities at resonance. We obtain existence theorems using Landesman-Lazer-type conditions and multiplicity theorems for problems with strong resonance at infinity. Our method of proof is based on the non-smooth critical point theory for locally Lipschitz functions and on a generalized version of the Ekeland variational principle.

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49

Sun, Juntao, and Tsung-fang Wu. "Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 2 (2016): 435–48. http://dx.doi.org/10.1017/s0308210515000475.

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We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

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50

Fiscella, Alessio, Patrizia Pucci, and Binlin Zhang. "p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities." Advances in Nonlinear Analysis 8, no. 1 (2018): 1111–31. http://dx.doi.org/10.1515/anona-2018-0033.

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Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

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