doi: 10.3934/dcds.2022068
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Positive solutions to a nonlinear fractional equation with an external source term

1. 

College of Science & Hubei Province Key Laboratory, of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430065, China

2. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

*Corresponding author: Shuangjie Peng

Received  June 2021 Revised  November 2021 Early access May 2022

Fund Project: The first author is supported by NNSF of China (No. 12071169). The third author is supported by NNSF of China (No. 11831009)

This paper deals with the following nonlinear fractional equation with an external source term
$ \begin{equation*} \label{eqS0.1} (-\Delta)^{s}u +u = K(x)u^{p}+f(x), \; u>0, \; x\in{\Bbb R}^N, \end{equation*} $
where
$ N>2s $
,
$ 0<s<1 $
,
$ 1<p<2_{\ast}(s)-1 $
,
$ 2_{\ast}(s) = \frac{2N}{N-2s} $
,
$ K(x) $
is a continuous function and
$ f\in L^{2}({\Bbb R}^{N})\cap L^{\infty}({\Bbb R}^{N}) $
. Using a Lyapunov-Schmidt reduction scheme, we prove that the equation admits
$ k $
-peak solutions for any integer
$ k>0 $
if
$ f $
is small and
$ K(x) $
satisfies some additional assumptions at infinity. The main difficulty is to improve the estimate of the remainder obtained in the reduction process.
Citation: Qi Li, Kefan Pan, Shuangjie Peng. Positive solutions to a nonlinear fractional equation with an external source term. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022068
References:
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A. AmbrosettiE. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

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V. Ambrosio and H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in ${\Bbb R}^N$, J. Dynam. Differ. Equ., 30 (2018), 1119-1143.  doi: 10.1007/s10884-017-9590-6.

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M. BhaktaS. Chakraborty and P. Pucci, Nonhomogeneous systems involving critical or subcritical nonlinearities, Differ. Integral Equ., 33 (2020), 323-336. 

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M. BhaktaS. Chakraborty and P. Pucci, Fractional Hardy-Sobolev equations with nonhomogeneous terms, Adv. Nonlinear Anal., 10 (2021), 1086-1116.  doi: 10.1515/anona-2020-0171.

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D. Bonheure and M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 675-688.  doi: 10.1016/j.anihpc.2008.06.002.

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J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Commun. Partial Differ. Equ., 29 (2004), 1877-1904.  doi: 10.1081/PDE-200040205.

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D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann., 336 (2006), 925-948.  doi: 10.1007/s00208-006-0021-y.

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

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D. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ${\Bbb R}^N$, Proc. R. Soc. Edinb. Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.

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T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.  doi: 10.1016/j.anihpc.2009.01.002.

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J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

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M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.

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E. Di NezzaG. 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.

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P. FelmerA. Quass and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[17]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in ${\Bbb R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[18]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[19]

T. Isernia, Positive solution for nonhomogeneous sublinear fractional equations in ${\Bbb R}^N$, Complex Var. Elliptic Equ., 63 (2018), 689-714.  doi: 10.1080/17476933.2017.1332052.

[20]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differ. Integral Equ., 10 (1997), 609-624. 

[21]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928. 

[22]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\Bbb R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[23]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[24]

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

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283.  doi: 10.1016/s0294-1449(16)30422-x.

[27]

W. Long and S. Peng, Positive vector solutions for a Schrödinger system with external sources terms, Nonlinear Differ. Equ. Appl., 27 (2020), Art. 5, 36 pp. doi: 10.1007/s00030-019-0608-0.

[28]

W. LongS. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discrete Contin. Dyn. Syst., 36 (2016), 917-939.  doi: 10.3934/dcds.2016.36.917.

[29]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.

[30]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452.  doi: 10.1112/S0024611598000148.

[31]

S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0.

[32]

J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in ${\Bbb R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.

[33]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 258 (2010), 3048-3081.  doi: 10.1016/j.jfa.2009.12.008.

[34]

X.-P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differ. Equ., 92 (1991), 163-178.  doi: 10.1016/0022-0396(91)90045-B.

show all references

References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u = a(x)u^p+f(x)$ in ${\Bbb R}^N$, Calc. Var. Partial Differ. Equ., 11 (2000), 63-95.  doi: 10.1007/s005260050003.

[2]

A. AmbrosettiE. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 30 (2007), 85-112.  doi: 10.1007/s00526-006-0079-0.

[3]

V. Ambrosio and H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in ${\Bbb R}^N$, J. Dynam. Differ. Equ., 30 (2018), 1119-1143.  doi: 10.1007/s10884-017-9590-6.

[4]

M. BhaktaS. Chakraborty and P. Pucci, Nonhomogeneous systems involving critical or subcritical nonlinearities, Differ. Integral Equ., 33 (2020), 323-336. 

[5]

M. BhaktaS. Chakraborty and P. Pucci, Fractional Hardy-Sobolev equations with nonhomogeneous terms, Adv. Nonlinear Anal., 10 (2021), 1086-1116.  doi: 10.1515/anona-2020-0171.

[6]

D. Bonheure and M. Ramos, Multiple critical points of perturbed symmetric strongly indefinite functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 675-688.  doi: 10.1016/j.anihpc.2008.06.002.

[7]

J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Commun. Partial Differ. Equ., 29 (2004), 1877-1904.  doi: 10.1081/PDE-200040205.

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[9]

D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann., 336 (2006), 925-948.  doi: 10.1007/s00208-006-0021-y.

[10]

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

[11]

D. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ${\Bbb R}^N$, Proc. R. Soc. Edinb. Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.

[12]

T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.  doi: 10.1016/j.anihpc.2009.01.002.

[13]

J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[14]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.

[15]

E. Di NezzaG. 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.

[16]

P. FelmerA. Quass and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[17]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in ${\Bbb R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.

[18]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[19]

T. Isernia, Positive solution for nonhomogeneous sublinear fractional equations in ${\Bbb R}^N$, Complex Var. Elliptic Equ., 63 (2018), 689-714.  doi: 10.1080/17476933.2017.1332052.

[20]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differ. Integral Equ., 10 (1997), 609-624. 

[21]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928. 

[22]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\Bbb R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[23]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[24]

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

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283.  doi: 10.1016/s0294-1449(16)30422-x.

[27]

W. Long and S. Peng, Positive vector solutions for a Schrödinger system with external sources terms, Nonlinear Differ. Equ. Appl., 27 (2020), Art. 5, 36 pp. doi: 10.1007/s00030-019-0608-0.

[28]

W. LongS. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discrete Contin. Dyn. Syst., 36 (2016), 917-939.  doi: 10.3934/dcds.2016.36.917.

[29]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2000), 213-227.  doi: 10.1112/S002461070000898X.

[30]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452.  doi: 10.1112/S0024611598000148.

[31]

S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0.

[32]

J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in ${\Bbb R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.

[33]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 258 (2010), 3048-3081.  doi: 10.1016/j.jfa.2009.12.008.

[34]

X.-P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differ. Equ., 92 (1991), 163-178.  doi: 10.1016/0022-0396(91)90045-B.

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