American Institute of Mathematical Sciences

April  2019, 12(2): 413-433. doi: 10.3934/dcdss.2019027

A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity

 1 College of Science, Civil Aviation University of China, Tianjin 300300, China 2 Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

* Corresponding author: Binlin Zhang

Dedicated to Vicenţiu D. Rădulescu on the occasion of his 60th birthday

Received  April 2017 Revised  January 2018 Published  August 2018

Fund Project: M. Xiang was supported by the National Natural Science Foundation of China (No. 11601515) and the Fundamental Research Funds for the Central Universities (No. 3122017080). B. Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667).

The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional
 $p$
-Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity:
 \begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}
where
 $M(t) = a+bt^{\theta-1}$
for
 $t\geq 0$
,
 $a\geq 0, b>0,\theta>1$
,
 $(-\Delta)_p^s$
is the fractional
 $p$
--Laplacian with
 $0 and $1
,
 $p_s^* = Np/(N-ps)$
is the critical Sobolev exponent,
 $\lambda>0$
is a parameter, and
 $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$
is a function. Under suitable assumptions on
 $f$
, we show that there exists
 $\lambda_0>0$
such that the above equation admits at least one nontrivial nonnegative solution provided
 $\lambda<\lambda_0$
by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any
 $k\in\mathbb{N}$
, there exists
 $\Lambda_k>0$
such that the above equation has
 $k$
pairs of nontrivial solutions if
 $\lambda<\Lambda_k$
. The main feature is that our paper covers the degenerate case, that is the coefficient of
 $(-\Delta)_p^s$
may be zero at zero. Moreover, the existence results are obtained when
 $f$
is discontinuous. Thus, our results are new even in the semilinear case.
Citation: Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027
References:
 [1] C. O. Alves and A. M. Bertone, A discontinuous problem involving the p-Laplacian operator and critical exponent in $\mathbb{R}^N$, Electron. J. Differential Equations, 2003 (2003), 1-10. [2] C. O. Alves, A. M. Bertone and J. V. Goncalves, A variational approach to discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 265 (2002), 103-127.  doi: 10.1006/jmaa.2001.7698. [3] D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [4] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014. [5] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016. [6] A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbb{R}^N$ perturbed by a local nonlinearity, Commun. Contemp. Math., 17 (2015), 1450039, 5 pp. doi: 10.1142/S0219199714500394. [7] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003. [8] L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremal functions for the fractional Sobolev inequality, Calc. Var. Partial Differentail Equations, 55 (2016), Art. 23, 32 pp. doi: 10.1007/s00526-016-0958-y. [9] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405. [11] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3. [12] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x. [13] J. Chabrowski, On multiple solutions for the non-homogeneous p-Laplacian with a critical Sobolev exponent, Differ. Integral Equations, 8 (1995), 705-716. [14] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [15] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [16] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [18] L. D'Onofrio, A. Fiscella and G. Molica Bisci, Perturbation methods for nonlocal Kirchhofftype problems, Fractional Calculus and Applied Analysis, 20 (2017), 829-853.  doi: 10.1515/fca-2017-0044. [19] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713.  doi: 10.1016/j.jmaa.2012.12.053. [20] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonliear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [21] A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004. [22] A. Fiscella, Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator, Differ. Integral Equations, 29 (2016), 513-530. [23] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC, Boca Raton, 2005. [24] J. V. Goncalves and C. O. Alves, Existence of positive solutions for m-Laplacian equations in $\mathbb{R}^N$ involving critical exponents, Nonlinear Anal., 32 (1998), 53-70.  doi: 10.1016/S0362-546X(97)00452-5. [25] Y. He, G. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214. [26] X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6. [27] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [28] S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbb{R}^N$, Nonlinear Anal., 81 (2013), 31-41.  doi: 10.1016/j.na.2012.12.003. [29] P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part Ⅰ, Rev. Mat. Iberoam., 1 (1985), 145–201. [Erratum in Part Ⅱ, Rev. Mat. Iberoam, 1 (1985), 45–121]. doi: 10.4171/RMI/6. [30] J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066. [31] J. Mawhin and M. Willem, Critical Point Theory and Hamilton Systems, Springer Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7. [32] X. Mingqi, G. Molica Bisci, G. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357. [33] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [34] G. Molica Bisci and D. Repovš, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.  doi: 10.1016/j.jmaa.2014.05.073. [35] G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5. [36] S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Br´ezis-Nirenberg problem for the fractional p–Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), Art. 105, 25 pp. doi: 10.1007/s00526-016-1035-2. [37] A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity, C. R. Math. Acad. Sci. Paris Ser. I, 352 (2014), 295-298.  doi: 10.1016/j.crma.2014.01.015. [38] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profle decomposition, and concentration compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y. [39] P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), 3-36.  doi: 10.5565/PUBLMAT6211801. [40] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam, 32 (2016), 1-22.  doi: 10.4171/RMI/879. [41] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5. [42] P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102. [43] X. Ros-Oston and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144. [44] R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Revista Matemática Complutense, 28 (2015), 655-676.  doi: 10.1007/s13163-015-0170-1. [45] X. D. Shang, Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 385 (2012), 1033-1043.  doi: 10.1016/j.jmaa.2011.07.029. [46] F. L. Wang and M. Q. Xiang, Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-11. [47] M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055. [48] M. Q. Xiang, B. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional p–Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp. doi: 10.1098/rspa.2015.0034. [49] M. Q. Xiang, B. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028. [50] M. Q. Xiang, B. L. Zhang and X. Zhang, A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in $\mathbb{R}^N$, Adv. Nonlinear Stud., 17 (2017), 611-640.  doi: 10.1515/ans-2016-6002.

show all references

Dedicated to Vicenţiu D. Rădulescu on the occasion of his 60th birthday

References:
 [1] C. O. Alves and A. M. Bertone, A discontinuous problem involving the p-Laplacian operator and critical exponent in $\mathbb{R}^N$, Electron. J. Differential Equations, 2003 (2003), 1-10. [2] C. O. Alves, A. M. Bertone and J. V. Goncalves, A variational approach to discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 265 (2002), 103-127.  doi: 10.1006/jmaa.2001.7698. [3] D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [4] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014. [5] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016. [6] A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbb{R}^N$ perturbed by a local nonlinearity, Commun. Contemp. Math., 17 (2015), 1450039, 5 pp. doi: 10.1142/S0219199714500394. [7] B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003. [8] L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremal functions for the fractional Sobolev inequality, Calc. Var. Partial Differentail Equations, 55 (2016), Art. 23, 32 pp. doi: 10.1007/s00526-016-0958-y. [9] L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813. [10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405. [11] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3. [12] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x. [13] J. Chabrowski, On multiple solutions for the non-homogeneous p-Laplacian with a critical Sobolev exponent, Differ. Integral Equations, 8 (1995), 705-716. [14] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [15] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [16] A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [18] L. D'Onofrio, A. Fiscella and G. Molica Bisci, Perturbation methods for nonlocal Kirchhofftype problems, Fractional Calculus and Applied Analysis, 20 (2017), 829-853.  doi: 10.1515/fca-2017-0044. [19] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713.  doi: 10.1016/j.jmaa.2012.12.053. [20] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonliear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [21] A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.  doi: 10.1016/j.nonrwa.2016.11.004. [22] A. Fiscella, Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator, Differ. Integral Equations, 29 (2016), 513-530. [23] L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC, Boca Raton, 2005. [24] J. V. Goncalves and C. O. Alves, Existence of positive solutions for m-Laplacian equations in $\mathbb{R}^N$ involving critical exponents, Nonlinear Anal., 32 (1998), 53-70.  doi: 10.1016/S0362-546X(97)00452-5. [25] Y. He, G. B. Li and S. J. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214. [26] X. M. He and W. M. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500.  doi: 10.1007/s10231-012-0286-6. [27] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [28] S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbb{R}^N$, Nonlinear Anal., 81 (2013), 31-41.  doi: 10.1016/j.na.2012.12.003. [29] P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part Ⅰ, Rev. Mat. Iberoam., 1 (1985), 145–201. [Erratum in Part Ⅱ, Rev. Mat. Iberoam, 1 (1985), 45–121]. doi: 10.4171/RMI/6. [30] J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.  doi: 10.1016/j.jmaa.2015.04.066. [31] J. Mawhin and M. Willem, Critical Point Theory and Hamilton Systems, Springer Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7. [32] X. Mingqi, G. Molica Bisci, G. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357. [33] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [34] G. Molica Bisci and D. Repovš, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.  doi: 10.1016/j.jmaa.2014.05.073. [35] G. Molica Bisci and V. Rădulescu, Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5. [36] S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Br´ezis-Nirenberg problem for the fractional p–Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), Art. 105, 25 pp. doi: 10.1007/s00526-016-1035-2. [37] A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity, C. R. Math. Acad. Sci. Paris Ser. I, 352 (2014), 295-298.  doi: 10.1016/j.crma.2014.01.015. [38] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profle decomposition, and concentration compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y. [39] P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), 3-36.  doi: 10.5565/PUBLMAT6211801. [40] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam, 32 (2016), 1-22.  doi: 10.4171/RMI/879. [41] P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5. [42] P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102. [43] X. Ros-Oston and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144. [44] R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Revista Matemática Complutense, 28 (2015), 655-676.  doi: 10.1007/s13163-015-0170-1. [45] X. D. Shang, Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl., 385 (2012), 1033-1043.  doi: 10.1016/j.jmaa.2011.07.029. [46] F. L. Wang and M. Q. Xiang, Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent, Electron. J. Differential Equations, 2016 (2016), 1-11. [47] M. Q. Xiang, B. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055. [48] M. Q. Xiang, B. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional p–Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp. doi: 10.1098/rspa.2015.0034. [49] M. Q. Xiang, B. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028. [50] M. Q. Xiang, B. L. Zhang and X. Zhang, A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in $\mathbb{R}^N$, Adv. Nonlinear Stud., 17 (2017), 611-640.  doi: 10.1515/ans-2016-6002.
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