# American Institute of Mathematical Sciences

September  2018, 17(5): 2085-2123. doi: 10.3934/cpaa.2018099

## Concentration phenomena for critical fractional Schrödinger systems

 Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino 'Carlo Bo', Piazza della Repubblica, 13 61029 Urbino (Pesaro e Urbino), Italy

Received  October 2017 Revised  November 2017 Published  April 2018

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system
 \left\{ \begin{array}{*{35}{l}} \begin{align} & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+V(x)u={{Q}_{u}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{u}}(u,v)\ \ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+W(x)v={{Q}_{v}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{v}}(u,v)\ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & u,v>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ in }{{\mathbb{R}}^{N}}, \\ \end{align} & \text{ } & \text{ } & {} \\\end{array} \right.
where
 $\varepsilon>0$
is a parameter,
 $s∈ (0, 1)$
,
 $N>2s$
,
 $(-Δ)^{s}$
is the fractional Laplacian operator,
 $V:\mathbb{R}^{N} \to \mathbb{R}$
and
 $W:\mathbb{R}^{N} \to \mathbb{R}$
are positive Hölder continuous potentials,
 $Q$
and
 $K$
are homogeneous
 $C^{2}$
-functions having subcritical and critical growth respectively.
We relate the number of solutions with the topology of the set where the potentials
 $V$
and
 $W$
attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.
Citation: Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
##### References:
 [1] C. O. Alves, Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150. [2] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787. [3] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758. [4] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723. [5] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. [6] C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457. [7] V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. [8] V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388). [9] V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370. [10] V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062. [11] V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370. [12] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645. [13] D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. xxx+460 pp. [14] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479. [15] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. [16] H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. [17] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. [18] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. [19] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. [20] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. [21] W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153. [22] J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. [23] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. [24] D. C. de Morais Filho and M. A. S. Souto, Systems of $p$ -Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553. [25] M. Del Pino and P. L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. [26] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. [27] S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230. [28] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. [29] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119. [30] L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103. [31] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. [32] G. M. Figueiredo and M. F. Furtado, Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333. [33] A. Fiscella and P. Pucci, $p$ -fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. [34] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. [35] X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp. [36] H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704. [37] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. [38] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. [39] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201. [40] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135. [41] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp. [42] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. [43] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. [44] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. [45] X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. [46] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. [47] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. [48] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. [49] S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924. [50] Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system, J. Math. Anal. Appl., 334 (2007), 1308-1325 [51] K. Wang and J. Wei, On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153. [52] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. [53] Z. Xia, B. Zhang and D. Repovs, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68.

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##### References:
 [1] C. O. Alves, Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150. [2] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787. [3] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758. [4] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723. [5] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. [6] C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457. [7] V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. [8] V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388). [9] V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370. [10] V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062. [11] V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370. [12] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645. [13] D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. xxx+460 pp. [14] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479. [15] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. [16] H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. [17] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. [18] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. [19] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. [20] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. [21] W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153. [22] J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. [23] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. [24] D. C. de Morais Filho and M. A. S. Souto, Systems of $p$ -Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553. [25] M. Del Pino and P. L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. [26] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. [27] S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230. [28] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. [29] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119. [30] L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103. [31] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. [32] G. M. Figueiredo and M. F. Furtado, Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333. [33] A. Fiscella and P. Pucci, $p$ -fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. [34] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. [35] X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp. [36] H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704. [37] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. [38] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. [39] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201. [40] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135. [41] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp. [42] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. [43] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. [44] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. [45] X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. [46] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. [47] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970. [48] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. [49] S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924. [50] Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system, J. Math. Anal. Appl., 334 (2007), 1308-1325 [51] K. Wang and J. Wei, On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153. [52] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. [53] Z. Xia, B. Zhang and D. Repovs, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68.
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