Positive solutions to a nonlinear fractional equation with an external source term

Qi Li; Kefan Pan; Shuangjie Peng

doi:10.3934/dcds.2022068

Article Contents

2022, Volume 42, Issue 10: 4669-4706. Doi: 10.3934/dcds.2022068

This issue Previous Article Next Article Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on

$ \mathit{\boldsymbol{\mathbb{R}^N}} $

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
^*Corresponding author: Shuangjie Peng

Received: June 2021

Revised: November 2021

Early access: May 2022

Published: October 2022

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35R11; Secondary: 35J20, 35J60.

Citation:

Full Text(HTML)

Related Papers

Cited by

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. Ambrosetti, E. 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. Bhakta, S. Chakraborty and P. Pucci, Nonhomogeneous systems involving critical or subcritical nonlinearities, Differ. Integral Equ., 33 (2020), 323-336.
[5]	M. Bhakta, S. 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. Dávila, M. 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 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.
[16]	P. Felmer, A. 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. Frank, E. 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. Long, S. 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.