American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration
  • DCDS Home
  • This Issue
  • Next Article
    Moduli for heteroclinic connections involving saddle-foci and periodic solutions
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

[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

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

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[2]

Minbo Yang, Jianjun Zhang, Yimin Zhang. Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025
[3]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[4]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883
[5]

Kazuhiro Kurata, Kotaro Morimoto. Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1443-1482. doi: 10.3934/cpaa.2008.7.1443
[6]

Jaeyoung Byeon, Louis Jeanjean. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 255-269. doi: 10.3934/dcds.2007.19.255
[7]

E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731
[8]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055
[9]

Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3087-3102. doi: 10.3934/dcds.2015.35.3087
[10]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[11]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[12]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[13]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393
[14]

Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413
[15]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[16]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[17]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[18]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
[19]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391
[20]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences