Multi-peak positive solutions for a fractional nonlinear elliptic equation

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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

Abstract
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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 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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