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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

Xudong Shang 1, and  Jihui Zhang 1,

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].
Keywords: Fractional elliptic equation, Lyapunov-Schmidt reduction, critical point., multi-peak solutions.
Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B3.
Citation: Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  doi: 10.1016/S0294-1449(99)80021-3.  Google Scholar

[3]

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

[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.  doi: 10.1080/03605300903346721.  Google Scholar

[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.  doi: 10.1080/03605302.2011.562954.  Google Scholar

[6]

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

[7]

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

[8]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359.  doi: 10.3934/cpaa.2014.13.2359.  Google Scholar

[9]

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

[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.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.4418/2013.68.1.15.  Google Scholar

[14]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

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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.  doi: 10.1007/s00220-014-1919-y.  Google Scholar

[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.  doi: 10.1017/S0308210511000746.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.   Google Scholar

[25]

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

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.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  doi: 10.1016/S0294-1449(99)80021-3.  Google Scholar

[3]

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

[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.  doi: 10.1080/03605300903346721.  Google Scholar

[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.  doi: 10.1080/03605302.2011.562954.  Google Scholar

[6]

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

[7]

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

[8]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359.  doi: 10.3934/cpaa.2014.13.2359.  Google Scholar

[9]

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

[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.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.4418/2013.68.1.15.  Google Scholar

[14]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[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.  doi: 10.1007/s00220-014-1919-y.  Google Scholar

[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.  doi: 10.1017/S0308210511000746.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.   Google Scholar

[25]

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

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