Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity

Qing Yang; Chuanzhi Bai; Qing Yang; Chuanzhi Bai

doi:10.3934/math.2021051

AIMS Mathematics

2021, Volume 6, Issue 1: 868-881. doi: 10.3934/math.2021051

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Research article

Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity

Qing Yang ,
Chuanzhi Bai ^,

School of Mathematical Science, Huaiyin Normal University, Huai'an, Jiangsu Province, 223300, China

Received: 20 September 2020 Accepted: 27 October 2020 Published: 03 November 2020
MSC : 35J20, 35J65, 35R11

In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity $ \left\{ \begin{array} {ll} \left(a+b \iint_{\Omega^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dxdy\right)(-\Delta)^s u + V(x)u = Q(x) |u|^{p-2}u \ln u^2, & {\rm in } \ \Omega, \\ u = 0, & {\rm in } \ \mathbb{R}^N \setminus \Omega, \end{array} \right. $ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, $V, Q$ are continuous, $V, Q \ge 0$. $a, b > 0$ are constants, $4 < p < 2_s^* : = \frac{2N}{N-2s}$. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.
- fractional Kirchhoff-Schrodinger-type equation,
- sign-changing solutions,
- logarithmic nonlinearity,
- variation methods
Citation: Qing Yang, Chuanzhi Bai. Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity[J]. AIMS Mathematics, 2021, 6(1): 868-881. doi: 10.3934/math.2021051

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Abstract

In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity $ \left\{ \begin{array} {ll} \left(a+b \iint_{\Omega^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dxdy\right)(-\Delta)^s u + V(x)u = Q(x) |u|^{p-2}u \ln u^2, & {\rm in } \ \Omega, \\ u = 0, & {\rm in } \ \mathbb{R}^N \setminus \Omega, \end{array} \right. $ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, $V, Q$ are continuous, $V, Q \ge 0$. $a, b > 0$ are constants, $4 < p < 2_s^* : = \frac{2N}{N-2s}$. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.

References

[1]	K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297. doi: 10.1134/S0202289310040067
[2]	P. D'Avenia, E. Montefusco, M. Squassina, On the logarithmic Schrodinger equation, Commun. Contemp. Math., 16 (2014), 313-402.
[3]	M. Squassina, A. Szulkin, Multiple solutions to logarithmic Schrodinger equations with periodic potential, Calc. Var., 54 (2015), 585-597. doi: 10.1007/s00526-014-0796-8
[4]	W. C. Troy, Uniqueness of positive ground state solutions of the logarithmic Schrodinger equation, Arch. Ration. Mech. Anal., 222 (2016), 1581-1600. doi: 10.1007/s00205-016-1028-5
[5]	S. Tian, Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity, J. Math. Anal. Appl., 454 (2017), 816-228. doi: 10.1016/j.jmaa.2017.05.015
[6]	C. Ji, A. Szulkin, A logarithmic Schrodinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254. doi: 10.1016/j.jmaa.2015.11.071
[7]	C. O. Alves, D. C. de Morais Filho, Existence and concentration of positive solutions for a Schrodinger logarithmic equation, Z. Angew. Math. Phys., 69 (2018), 144.
[8]	S. Chen, X. H. Tang, Ground state sign-changing solutions for elliptic equations with logarithmic nonlinearity, Acta Math. Hungar., 157 (2019), 27-38. doi: 10.1007/s10474-018-0891-y
[9]	A. Fiscela, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011
[10]	Y. Li, D. Wang, J. Zhang, Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity, AIMS Mathematics, 5 (2020), 2100-2112. doi: 10.3934/math.2020139
[11]	L. Wen, X. H. Tang, S. Chen, Ground state sign-changing solutions for Kirchhoff equations with logarithmic nonlinearity, Electron. J. Qual. Theor., 47 (2019), 1-13.
[12]	W. Shuai, Multiple solutions for logarithmic Schrodinger equations, Nonlinearity, 32 (2019), 2201-2225. doi: 10.1088/1361-6544/ab08f4
[13]	R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discret. Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105
[14]	Q. Zhou, K. Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian, Fract. Calc. Appl. Anal., 18 (2015), 133-145.
[15]	D. Yang, C. Bai, Multiplicity of weak positive solutions for fractional p & q Laplacian problem with singular nonlinearity, J. Funct. Space., 2020 (2020), 1-8.
[16]	G. M. Bisci, V. D. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, Cambridge: Cambridge University Press, 2016.
[17]	E. Cinti, F. Colasuonno, A nonlocal supercritical Neumann problem, J. Differ. Equations, 268 (2020), 2246-2279. doi: 10.1016/j.jde.2019.09.014
[18]	V. Ambrosio, J. Mawhin, G. M. Bisci, (Super)Critical nonlocal equations with periodic boundary conditions, Sel. Math. New Ser., 24 (2018), 3723-3751. doi: 10.1007/s00029-018-0398-y
[19]	C. O. Alves, C. E. T. Ledesma, Fractional elliptic problem in exterior domains with nonlocal Neumann condition, Nonlinear Anal., 195 (2020), 111732. doi: 10.1016/j.na.2019.111732
[20]	T. Isernia, Sign-changing solutions for a fractional Kirchhoff equation, Nonlinear Anal., 190 (2020), 111623. doi: 10.1016/j.na.2019.111623
[21]	L. M. Del Pezzo, A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, J. Differ. Equations, 263 (2017), 765-778. doi: 10.1016/j.jde.2017.02.051
[22]	X. Shang, J. Zhang, Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Commun. Pure Appl. Anal., 13 (2014), 567-584.
[23]	C. Miranda, Un'osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7.
[24]	M. Willem, Minimax theorems, Bosten: Birkhauser, 1996.
[25]	K. Deimling, Nonlinear functional analysis, Berlin: Springer-Verlag, 1985.
[26]	D. Guo, Nonlinear functional analysis, Beijing: Higher Education Press, 2015.

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