Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity

Mengyu Wang; Xinmin Qu; Huiqin Lu; Mengyu Wang; Xinmin Qu; Huiqin Lu

doi:10.3934/math.2021297

AIMS Mathematics

2021, Volume 6, Issue 5: 5028-5039. doi: 10.3934/math.2021297

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Research article

Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity

School of Mathematics Statistics, Shandong Normal University, Jinan, 250358, PR China

Received: 26 December 2020 Accepted: 01 March 2021 Published: 08 March 2021
MSC : 35J65, 47J05, 47J30

In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth.
- fractional Laplacian equation,
- sign-changing solution,
- Ekeland's variational principle,
- Brouwer's degree theory,
- variational methods
Citation: Mengyu Wang, Xinmin Qu, Huiqin Lu. Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity[J]. AIMS Mathematics, 2021, 6(5): 5028-5039. doi: 10.3934/math.2021297

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Abstract

In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth.

References

[1]	D. Applebaum, L$\acute{e}$vy processes and stochastic calculus, In: Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press, 2009.
[2]	R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887–898. doi: 10.1016/j.jmaa.2011.12.032
[3]	R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont. Dyn. Syst., 33 (2013), 2105–2137. doi: 10.3934/dcds.2013.33.2105
[4]	R. Servadei, E. Valdinoci, Fractional Laplacian equations with critivcal Sobolev exponent, Rev. Mat. Complut., 28 (2015), 655–676. doi: 10.1007/s13163-015-0170-1
[5]	M. Mu, H. Lu, Existence and multiplicity of positive solutions for schrodinger-kirchhoff-poisson system with singularity, J. Funct. Space., 2017 (2017), 1–12.
[6]	B. Yan, D. Wang, The multiplicity of positive solutions for a class of nonlocal elliptic problem, J. Math. Anal. Appl., 442 (2016), 72–102. doi: 10.1016/j.jmaa.2016.04.023
[7]	H. Lu, Multiple positive solutions for singular semipositone periodic boundary value problems with derivative dependence, J. Appl. Math., 2012 (2012), 857–868.
[8]	Y. Wang, Y. Liu, Y. Cui, Multiple solutions for a nonlinear fractional boundary value problem via critical point theory, J. Funct. Space., 2017 (2017), 1–8.
[9]	A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275–1287. doi: 10.1016/j.na.2008.02.011
[10]	Z. Liu, F. A. Van Heerden, A. Francois, Z. Wang, Nodal type bound states of Schr$\ddot{o}$dinger equations via invariant set and minimax methods, J. Differ. Equ., 214 (2005), 358–390. doi: 10.1016/j.jde.2004.08.023
[11]	Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456–463. doi: 10.1016/j.jmaa.2005.06.102
[12]	H. Lu, X. Zhang, Positive solution for a class of nonlocal elliptic equations, Appl. Math. Lett., 88 (2019), 125–131. doi: 10.1016/j.aml.2018.08.019
[13]	Z. Liu, Z. Wang, J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schr$\ddot{o}$dinger-Poisson system, Ann. Mat. Pur. Appl., 195 (2016), 775–794. doi: 10.1007/s10231-015-0489-8
[14]	Y. Wang, Y. Liu, Y. Cui, Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations, Bound. Value Probl., 2018 (2018), 1–21. doi: 10.1186/s13661-017-0918-2
[15]	Z. Liu, J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differ. Equ., 172 (2001), 257–299. doi: 10.1006/jdeq.2000.3867
[16]	B. Yan, C. An, The sign-changing solutions for a class of nonlocal elliptic problem in an annulus, Topol. Method. Nonlinear Anal., 55 (2020), 1–18.
[17]	H. Lu, Y. Wang, Y. Liu, Nodal solutions for some second-order semipositone integral boundary value problems, Abstr. Appl. Anal., 2014 (2014), 1–6.
[18]	F. Jin, B. Yan, The sign-changing solutions for nonlinear elliptic problem with Carrier type, J. Math. Anal. Appl., 487 (2020), 1–24,
[19]	H. Luo, Sign-changing solutions for non-local elliptic equations, Electron. J. Differ. Equ., 180 (2017), 1–15.
[20]	H. Lu, X. Qu, J. Wang, Sign-changing and constant-sign solutions for elliptic problems involving nonlocal integro-differential operators, SN Partial Differ. Equ. Appl., 1 (2020), 1–17. doi: 10.1007/s42985-019-0002-0

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