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Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations
Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
1. | School of Science, Jiangxi University of Science and Technology, Ganzhou, Jiangxi 341000, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China |
$\left\{ \begin{align} &{{\varepsilon }^{2\alpha }}{{\left( -\Delta \right)}^{a}}u+V\left( x \right)u=f\left( u \right),\ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{a}}\left( {{\mathbb{R}}^{N}} \right),u>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ \end{align} \right.$ |
$0<α<1$ |
$N>2α$ |
$\varepsilon>0$ |
$V$ |
$f$ |
$\text{cat}_{M_{δ}}(M)$ |
References:
[1] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.
doi: 10.1080/03605302.2011.593013. |
[2] |
A. Ambrosetti and A. Malchiodi,
Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618260. |
[3] |
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. |
[4] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational. Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[5] |
C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities, preprint, arXiv: 1609.01911. |
[6] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
doi: 10.1007/978-3-319-28739-3. |
[7] |
J. Byeon, O. Kwon and J. Seok,
Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.
|
[8] |
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.
doi: 10.1016/j.anihpc.2013.02.001. |
[9] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[10] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
G. Chen,
Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 927-949.
doi: 10.1088/0951-7715/28/4/927. |
[12] |
G. Chen and Y. Zheng,
Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[13] |
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507.
doi: 10.1063/1.3701574. |
[14] |
S. Cingolani and M. Lazzo,
Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[15] |
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.
doi: 10.2140/apde.2015.8.1165. |
[16] |
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. |
[17] |
M. Del Pino and P. Felmer,
Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[18] |
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. |
[19] |
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. (2016),
doi: 10.1007/s00229-016-0878-3. |
[20] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$ in Appunti. Edizioni della Normale Scuola Normale di Pisa (2017). arXiv: 1506.01748.
doi: 10.1007/978-88-7642-601-8. |
[21] |
M. M. Fall and E. Valdinoci,
Uniqueness and nondegeneracy of positive solutions of $(Δ)^s u+u=u^p$ in $\mathbb{R}^N$ when $s$ is close to $1$, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[22] |
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. |
[23] |
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.
doi: 10.1017/S0308210511000746. |
[24] |
G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrdinger equation in $\mathbb{R}^N$
Nonlinear Differ. Equ. Appl. 23 (2016), 12.
doi: 10.1007/s00030-016-0355-4. |
[25] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-53.
doi: 10.5186/aasfm.2015.4009. |
[26] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[27] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[28] |
N. Laskin,
Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.
|
[29] |
W. Liu, Multiple solutions for a fractional nonlinear Schrödinger equation with a general nonlinearity, prepared. |
[30] |
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. |
[31] |
R. Servadei and E. Valdinoci,
Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.
doi: 10.4171/RMI/750. |
[32] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[33] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[34] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[35] |
X. Shang, J. Zhang and Y. Yang, On fractional Schödinger equation in $\mathbb{R}^N$ with critical growth J. Math. Phys. 54 (2013), 121502.
doi: 10.1063/1.4835355. |
[36] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[37] |
X. Shang and J. Zhang,
Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.
doi: 10.1016/j.jde.2014.10.012. |
show all references
References:
[1] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.
doi: 10.1080/03605302.2011.593013. |
[2] |
A. Ambrosetti and A. Malchiodi,
Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618260. |
[3] |
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. |
[4] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational. Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[5] |
C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities, preprint, arXiv: 1609.01911. |
[6] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
doi: 10.1007/978-3-319-28739-3. |
[7] |
J. Byeon, O. Kwon and J. Seok,
Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.
|
[8] |
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.
doi: 10.1016/j.anihpc.2013.02.001. |
[9] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[10] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
G. Chen,
Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 927-949.
doi: 10.1088/0951-7715/28/4/927. |
[12] |
G. Chen and Y. Zheng,
Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[13] |
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507.
doi: 10.1063/1.3701574. |
[14] |
S. Cingolani and M. Lazzo,
Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[15] |
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.
doi: 10.2140/apde.2015.8.1165. |
[16] |
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. |
[17] |
M. Del Pino and P. Felmer,
Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[18] |
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. |
[19] |
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. (2016),
doi: 10.1007/s00229-016-0878-3. |
[20] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$ in Appunti. Edizioni della Normale Scuola Normale di Pisa (2017). arXiv: 1506.01748.
doi: 10.1007/978-88-7642-601-8. |
[21] |
M. M. Fall and E. Valdinoci,
Uniqueness and nondegeneracy of positive solutions of $(Δ)^s u+u=u^p$ in $\mathbb{R}^N$ when $s$ is close to $1$, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[22] |
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. |
[23] |
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.
doi: 10.1017/S0308210511000746. |
[24] |
G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrdinger equation in $\mathbb{R}^N$
Nonlinear Differ. Equ. Appl. 23 (2016), 12.
doi: 10.1007/s00030-016-0355-4. |
[25] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-53.
doi: 10.5186/aasfm.2015.4009. |
[26] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[27] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[28] |
N. Laskin,
Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.
|
[29] |
W. Liu, Multiple solutions for a fractional nonlinear Schrödinger equation with a general nonlinearity, prepared. |
[30] |
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. |
[31] |
R. Servadei and E. Valdinoci,
Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.
doi: 10.4171/RMI/750. |
[32] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[33] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[34] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[35] |
X. Shang, J. Zhang and Y. Yang, On fractional Schödinger equation in $\mathbb{R}^N$ with critical growth J. Math. Phys. 54 (2013), 121502.
doi: 10.1063/1.4835355. |
[36] |
X. Shang and J. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[37] |
X. Shang and J. Zhang,
Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.
doi: 10.1016/j.jde.2014.10.012. |
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