Existence and multiplicity of positive solutions for one-dimensional $ p $-Laplacian problem with sign-changing weight

Liangying Miao; Man Xu; Zhiqian He; Liangying Miao; Man Xu; Zhiqian He

doi:10.3934/era.2023156

Electronic Research Archive

2023, Volume 31, Issue 6: 3086-3096. doi: 10.3934/era.2023156

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Existence and multiplicity of positive solutions for one-dimensional $ p $-Laplacian problem with sign-changing weight

1.
College of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
3.
School of Mathematics and Physics, Qinghai University, Xining 810016, China

Academic Editor: Fengqi Yi

Received: 15 January 2023 Revised: 15 February 2023 Accepted: 15 March 2023 Published: 23 March 2023

In this paper, we show the positive solutions set for one-dimensional $ p $-Laplacian problem with sign-changing weight contains a reversed $ S $-shaped continuum. By figuring the shape of unbounded continuum of positive solutions, we identify the interval of bifurcation parameter in which the $ p $-Laplacian problem has one or two or three positive solutions according to the asymptotic behavior of nonlinear term at 0 and $ \infty $. The proof of the main result is based upon bifurcation technique.
- $ p $-Laplacian problem,
- positive solution,
- multiplicity,
- bifurcation,
- sign-changing weight
Citation: Liangying Miao, Man Xu, Zhiqian He. Existence and multiplicity of positive solutions for one-dimensional $ p $-Laplacian problem with sign-changing weight[J]. Electronic Research Archive, 2023, 31(6): 3086-3096. doi: 10.3934/era.2023156

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Abstract

In this paper, we show the positive solutions set for one-dimensional $ p $-Laplacian problem with sign-changing weight contains a reversed $ S $-shaped continuum. By figuring the shape of unbounded continuum of positive solutions, we identify the interval of bifurcation parameter in which the $ p $-Laplacian problem has one or two or three positive solutions according to the asymptotic behavior of nonlinear term at 0 and $ \infty $. The proof of the main result is based upon bifurcation technique.

References

[1]	M. del Pino, M. Elgueta, R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(\|u'\|^{p-2}u')'+f(t, u) = 0, u(0) = u(T) = 0, p>1$, J. Differ. Equations, 80 (1989), 1–13. https://doi.org/10.1016/0022-0396(89)90093-4 doi: 10.1016/0022-0396(89)90093-4
[2]	Y. H. Lee, I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differ. Equations, 229 (2006), 229–256. https://doi.org/10.1016/j.jde.2006.03.021 doi: 10.1016/j.jde.2006.03.021
[3]	Y. H. Lee, S. U. Kim, E. K. Lee, Three solutions theorem for $p$-Laplacian problems with a singular weight and its application, Abstr. Appl. Anal., 2014 (2014), 1–9. https://doi.org/10.1155/2014/502756 doi: 10.1155/2014/502756
[4]	G. Dai, R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differ. Equations, 252 (2012), 2448–2468. https://doi.org/10.1016/j.jde.2011.09.026 doi: 10.1016/j.jde.2011.09.026
[5]	G. Dai, Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323–5345. https://doi.org/10.1016/j.jde.2004.10.005 doi: 10.1016/j.jde.2004.10.005
[6]	M. del Pino, R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J. Differ. Equations, 92 (1991), 226–251. https://doi.org/10.1016/0022-0396(91)90048-E doi: 10.1016/0022-0396(91)90048-E
[7]	F. Ye, X. Han, Global bifurcation for $N$-dimensional $p$-Laplacian problem and its applications, Complex Var. Elliptic Equations, 67 (2022), 3074–3091. https://doi.org/10.1080/17476933.2021.1984437 doi: 10.1080/17476933.2021.1984437
[8]	P. Drábek, Y. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbb{R}^N$, Trans. Am. Math. Soc., 349 (1997), 171–188. https://doi.org/10.1090/S0002-9947-97-01788-1 doi: 10.1090/S0002-9947-97-01788-1
[9]	R. Ma, X. Liu, J. Xu, Nodal solutions of the $p$-Laplacian with sign-changing weight, Abstr. Appl. Anal., 2013 (2015), 8. https://doi.org/10.1155/2013/406350 doi: 10.1155/2013/406350
[10]	G. Dai, X. Han, R. Ma, Unilateral global bifurcation and nodal solutions for the $p$-Laplacian with sign-changing weight, Complex Var. Elliptic Equations, 59 (2014), 847–862. https://doi.org/10.1080/17476933.2013.791686 doi: 10.1080/17476933.2013.791686
[11]	I. Sim, S. Tanaka, Three positive solutions for one-dimensional $p$-Laplacian problem with sign-changing weight, Appl. Math. Lett., 49 (2015), 42–50. https://doi.org/10.1016/j.aml.2015.04.007 doi: 10.1016/j.aml.2015.04.007
[12]	T. Chen, R. Ma, Three positive solutions of $N$-dimensional $p$-Laplacian with indefinite weight, (2019), 1–14. https://doi.org/10.14232/EJQTDE.2019.1.19
[13]	T. Kusano, M. Naito, Sturm-Liouville eigenvalue problems from half-linear ordinary differential equations, Rocky Mt. J. Math., 31 (2001), 1039–1054. https://doi.org/10.1216/rmjm/1020171678 doi: 10.1216/rmjm/1020171678
[14]	J. L. Díaz, Non-Lipschitz heterogeneous reaction with a $p$-Laplacian operator, AIMS Math., 7 (2022), 3395–3417. https://doi.org/10.3934/math.2022189 doi: 10.3934/math.2022189
[15]	Y. Yang, Q. Wang, Multiple positive solutions for one-dimensional third order $p$-Laplacian equations with integral boundary conditions, Acta Math. Appl. Sin. Engl. Ser., 38 (2022), https://doi.org/10.1007/s10255-022-1065-9 doi: 10.1007/s10255-022-1065-9
[16]	Y. Ning, D. Lu, A. Mao, Existence and subharmonicity of solutions for nonsmooth $p$-Laplacian systems, AIMS Math., 6 (2021), 10947–10963. https://doi.org/10.3934/math.2021636 doi: 10.3934/math.2021636
[17]	F. Zeng, P. Shi, M. Jiang, Global existence and finite time blow-up for a class of fractional $p$-Laplacian Kirchhoff type equations with logarithmic nonlinearity, AIMS Math., 6 (2021), 2559–2578. https://doi.org/10.3934/math.2021155 doi: 10.3934/math.2021155
[18]	V. A. Galaktionov, Three types of self-similar blow-up for the fourth-order $p$-Laplacian equation with source, J. Comput. Appl. Math., 223 (2009), 326–355. https://doi.org/10.1016/j.cam.2008.01.027 doi: 10.1016/j.cam.2008.01.027
[19]	J. D. Palencia, A. Otero, Oscillatory solutions and smoothing of a higher-order $p$-Laplacian operator, Electron. Res. Arch., 30 (2022), 3527–3547. https://doi.org/10.3934/era.2022180 doi: 10.3934/era.2022180
[20]	S. Rahman, J. D. Palencia, Regularity and analysis of solutions for a MHD flow with a $p$-Laplacian operator and a generalized Darcy-Forchheimer term, Eur. Phys. J. Plus, 137 (2022), 1–16. https://doi.org/10.1140/epjp/s13360-022-03555-0 doi: 10.1140/epjp/s13360-022-03555-0
[21]	R. Ma, Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364–4376. https://doi.org/10.1016/j.na.2009.02.113 doi: 10.1016/j.na.2009.02.113

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