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Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation
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 |
$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$ |
$\varepsilon$ |
$(-Δ)^{s}$ |
$s ∈ (0,1)$ |
$N> 2s$ |
$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$ |
$\text{inf}_{\mathbb{R}^{N}} V(x)>0$ |
$k$ |
$x^{j} ∈ \mathbb{R}^{N}$ |
$j = 1,···,k$ |
$V(x^{j})$ |
$f$ |
$k$ |
$\varepsilon>0$ |
$\varepsilon$ |
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] |
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. |
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