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July  2015, 35(7): 3183-3201. doi: 10.3934/dcds.2015.35.3183

Multi-peak positive solutions for a fractional nonlinear elliptic equation

1. 

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

Received  September 2014 Revised  November 2014 Published  January 2015

In this paper we study the existence of positive multi-peak solutions to the semilinear equation \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N}) \end{eqnarray*} where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in (0,1)$, $\varepsilon$ is a positive small parameter, $2 < p < \frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function. For any positive integer $k$, we prove the existence of a positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].
Citation: Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
References:
[1]

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.

[2]

D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré, 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3.

[3]

D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264. doi: 10.1017/S030821050002134X.

[4]

D. Cao and S. Peng, Semi-classical bound states for Schröinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591. doi: 10.1080/03605300903346721.

[5]

A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954.

[6]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992. doi: 10.1016/j.jde.2014.01.027.

[7]

G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations,, , (). 

[8]

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.

[9]

M. Cheng, Bound state for the fractional Schrödinger equations with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp. doi: 10.1063/1.3701574.

[10]

W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029.

[11]

J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum,, , (). 

[12]

J. Dávila, M. Del Pino and J. 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.

[13]

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.

[14]

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.

[15]

M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $ (-\Delta)^s u + u = u^p$ in $\mathbbR^N$ when $s$ close to 1, Comm. Math. Phys., 329 (2014), 383-404. doi: 10.1007/s00220-014-1919-y.

[16]

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

[17]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , (). 

[18]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[19]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E., 62 (2000), p. 3135. doi: 10.1103/PhysRevE.62.3135.

[20]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689. doi: 10.1512/iumj.2009.58.3611.

[21]

W. Long, S. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schrödinger equations,, , (). 

[22]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2002), 213-227. doi: 10.1112/S002461070000898X.

[23]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 031501, 17 pp. doi: 10.1063/1.4793990.

[24]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vec e$MA, 49 (2009), 33-44.

[25]

L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, , (). 

show all references

References:
[1]

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.

[2]

D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré, 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3.

[3]

D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264. doi: 10.1017/S030821050002134X.

[4]

D. Cao and S. Peng, Semi-classical bound states for Schröinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591. doi: 10.1080/03605300903346721.

[5]

A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954.

[6]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992. doi: 10.1016/j.jde.2014.01.027.

[7]

G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations,, , (). 

[8]

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.

[9]

M. Cheng, Bound state for the fractional Schrödinger equations with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp. doi: 10.1063/1.3701574.

[10]

W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029.

[11]

J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum,, , (). 

[12]

J. Dávila, M. Del Pino and J. 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.

[13]

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.

[14]

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.

[15]

M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $ (-\Delta)^s u + u = u^p$ in $\mathbbR^N$ when $s$ close to 1, Comm. Math. Phys., 329 (2014), 383-404. doi: 10.1007/s00220-014-1919-y.

[16]

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

[17]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , (). 

[18]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[19]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E., 62 (2000), p. 3135. doi: 10.1103/PhysRevE.62.3135.

[20]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689. doi: 10.1512/iumj.2009.58.3611.

[21]

W. Long, S. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schrödinger equations,, , (). 

[22]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2002), 213-227. doi: 10.1112/S002461070000898X.

[23]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 031501, 17 pp. doi: 10.1063/1.4793990.

[24]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vec e$MA, 49 (2009), 33-44.

[25]

L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, , (). 

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