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November  2018, 17(6): 2239-2259. doi: 10.3934/cpaa.2018107

Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Xudong Shang 1,, and  Jihui Zhang 2,

1. 

School of Mathematics, Nanjing Normal University Taizhou College, Taizhou 225300, China
2. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

* Corresponding author

Received  April 2017 Revised  September 2017 Published  June 2018

In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation
$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$
where
$\varepsilon$
 is a positive parameter,
$(-Δ)^{s}$
 is the fractional Laplacian,
$s ∈ (0,1)$
 and
$N> 2s$
. Suppose that the potential
$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$
 satisfies
$\text{inf}_{\mathbb{R}^{N}} V(x)>0$
, and there exist
$k$
 points
$x^{j} ∈ \mathbb{R}^{N}$
 such that for each
$j = 1,···,k$
,
$V(x^{j})$
 are strict global minimum. When
$f$
 is subcritical, we prove that the problem has at least
$k$
 positive solutions for
$\varepsilon>0$
 small. Moreover, we establish the concentration property of the solutions as
$\varepsilon$
 tends to zero.
Keywords: Fractional Schrödinger equations, Multiplicity of solutions, Ekeland's variational principle.
Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.
Citation: Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
References:
[1]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.

[2]

K-C Chang, Methods in nonlinear analysis, Springer-verlag Berlin Heidelberg, 2005.

[3]

D. Cao and E. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $ \mathbb{R}^{N} $, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.

[4]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[5]

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.  doi: 10.1007/s00229-016-0878-3.

[6]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for nonlocal equtions with Dirichlet datum, Anal. PDE., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.

[7]

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.

[8]

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.

[9]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[10]

J. Dávila, M. Del Pino and J. C. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[11]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[12]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[13]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, (2016), 55-91.  doi: 10.1007/s00526-016-1045-0.

[14]

N. Laskin, Fractional Schrödinger equation, Phy. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Analy. Non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.

[16]

R. Pabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^{N} $, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990.

[18]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996.

show all references

References:
[1]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.

[2]

K-C Chang, Methods in nonlinear analysis, Springer-verlag Berlin Heidelberg, 2005.

[3]

D. Cao and E. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $ \mathbb{R}^{N} $, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.

[4]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[5]

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.  doi: 10.1007/s00229-016-0878-3.

[6]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for nonlocal equtions with Dirichlet datum, Anal. PDE., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.

[7]

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.

[8]

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.

[9]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15.

[10]

J. Dávila, M. Del Pino and J. C. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[11]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.

[12]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[13]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, (2016), 55-91.  doi: 10.1007/s00526-016-1045-0.

[14]

N. Laskin, Fractional Schrödinger equation, Phy. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Analy. Non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.

[16]

R. Pabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^{N} $, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990.

[18]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996.

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