This paper considers a singular Kirchhoff equation with convection and a parameter. By defining new sub-supersolutions, we prove a new sub-supersolution theorem. Combining method of sub-supersolution with the comparison principle, for Kirchhoff equation with convection, we get the conclusion about positive solutions when nonlinear term is singular and sign-changing.
Citation: Xiaohui Qiu, Baoqiang Yan. The existence and nonexistence of positive solutions for a singular Kirchhoff equation with convection term[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 10581-10601. doi: 10.3934/mbe.2022494
This paper considers a singular Kirchhoff equation with convection and a parameter. By defining new sub-supersolutions, we prove a new sub-supersolution theorem. Combining method of sub-supersolution with the comparison principle, for Kirchhoff equation with convection, we get the conclusion about positive solutions when nonlinear term is singular and sign-changing.
[1] | M. Ghergu, V. R$\breve{a}$dulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635–646. https://doi.org/10.1016/j.jmaa.2005.03.012 doi: 10.1016/j.jmaa.2005.03.012 |
[2] | H. Cheng, R. Yuan, Traveling waves of some Holling-Tanner predator-prey system with nonlocal diffusion, Appl. Math. Comput., 338 (2018), 12–24. https://doi.org/10.1016/j.amc.2018.04.049 doi: 10.1016/j.amc.2018.04.049 |
[3] | H. Cheng, R. Yuan, Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 5433–5454. https://doi.org/10.3934/dcds.2017236 doi: 10.3934/dcds.2017236 |
[4] | H. Cheng, R. Yuan, Traveling waves of a nonlocal dispersal Kermack-McKerndrick epidemic model with delayed transmission, J. Evol. Equations, 17 (2017), 979–1002. https://doi.org/10.1007/s00028-016-0362-2 doi: 10.1007/s00028-016-0362-2 |
[5] | Y. Liu, Y. Zheng, H. Li, F. E. Alsaadi, B. Ahmad, Control design for output tracking of delayed Boolean control networks, J. Comput. Appl. Math., 327 (2018), 188–195. https://doi.org/10.1016/j.cam.2017.06.016 doi: 10.1016/j.cam.2017.06.016 |
[6] | Y. Liu, Bifurcation techniques for a class of boundary value problemsof fractional impulsive differential equations, J. Nonlinear Sci. Appl., 8 (2015), 340–353. http://dx.doi.org/10.22436/jnsa.008.04.07 doi: 10.22436/jnsa.008.04.07 |
[7] | Y. Liu, D. O'Regan, Controllability of impulsive functional differential systems with nonlocal conditions, Electron. J. Differ. Equations., 194 (2013), 1–10. https://doi.org/10.1016/j.amc.2011.01.107 doi: 10.1016/j.amc.2011.01.107 |
[8] | J. Xu, J. Jiang, D. O'Regan, Positive Solutions for a Class of $p$-Laplacian Hadamard Fractional-Order Three-Point, Boundary Value Probl., 8 (2020), 308. https://doi.org/10.3390/math8030308 doi: 10.3390/math8030308 |
[9] | C. O. Alves, F. J. S. A. Corrêa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85–93. https://doi.org/10.1016/j.camwa.2005.01.008 doi: 10.1016/j.camwa.2005.01.008 |
[10] | A. Bensedik, M. Bouchekif, On an elliptic equation of Kirchhoff type with a potential asymptotically linear at infinity, Math. Comput. Model., 49 (2009), 1089–1096. https://doi.org/10.1016/j.mcm.2008.07.032 doi: 10.1016/j.mcm.2008.07.032 |
[11] | B. Cheng, X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883–4892. https://doi.org/10.1016/j.na.2009.03.065 doi: 10.1016/j.na.2009.03.065 |
[12] | G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. |
[13] | T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967–1977. https://doi.org/10.1016/j.na.2005.03.021 doi: 10.1016/j.na.2005.03.021 |
[14] | A. Mao, S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239–243. https://doi.org/10.1016/j.jmaa.2011.05.021 doi: 10.1016/j.jmaa.2011.05.021 |
[15] | A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275–1287. https://doi.org/10.1016/j.na.2008.02.011 doi: 10.1016/j.na.2008.02.011 |
[16] | K. Perera, Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equations, 221, (2006), 246–255. https://doi.org/10.1016/j.jde.2005.03.006 |
[17] | J. J. Sun, C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212–1222. https://doi.org/10.1016/j.na.2010.09.061 doi: 10.1016/j.na.2010.09.061 |
[18] | Y. Wang, Y. Liu, Y. Cui, Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations, Boundary Value Probl., 193 (2018). https://doi.org/10.1186/s13661-018-1114-8 |
[19] | M. H. Yang, Z. Q. Han, Existence and multiplicity results for Kirchhoff type problems with four-superlinear potentials, Appl. Anal, 91 (2012), 2045–2055. https://doi.org/10.1080/00036811.2011.587808 doi: 10.1080/00036811.2011.587808 |
[20] | Y. Yang, J. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377–380. https://doi.org/10.1016/j.aml.2009.11.001 doi: 10.1016/j.aml.2009.11.001 |
[21] | 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. https://doi.org/10.1016/j.jmaa.2005.06.102 doi: 10.1016/j.jmaa.2005.06.102 |
[22] | C. O. Alves, F. J. S. A. Corrêa, A sub-supersolution approach for a quasilinear Kirchhoff equation, J. Math. Phys., 56 (2015), 591–608. https://doi.org/10.1063/1.4919670 doi: 10.1063/1.4919670 |
[23] | C. O. Alves, F. J. S. A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal., 8 (2014), 43–56. |
[24] | B. Yan, D. O'Regen, R. P. Agarwal, The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method, An. Sţ. Univ. Ovidius Constanţa, 26 (2018), 5–41. https://doi.org/10.2478/auom-2018-0001 |
[25] | C. O. Alves, D. P. Covei, Existence of solution for a class of nonlocal elliptic problem via sub-supersolution method, Nonlinear Anal. Real World Appl., 23 (2015), 1–8. https://doi.org/10.1016/j.nonrwa.2014.11.003 doi: 10.1016/j.nonrwa.2014.11.003 |
[26] | C. O. Alves, F. J. S. A. Corrêa, On the existence of positive solution for a class of singular systems involving quasilinear operators, Appl. Math. Comput., 185 (2007), 727–736. https://doi.org/10.1016/j.amc.2006.07.080 doi: 10.1016/j.amc.2006.07.080 |
[27] | S. Cui, Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems, Nonlinear Anal. 41 (2000), 149–176. https://doi.org/10.1016/S0362-546X(98)00271-5 |
[28] | X. Li, S. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE Trans. Autom. Control, 62 (2017), 406–411. https://doi.org/10.1109/TAC.2016.2530041 doi: 10.1109/TAC.2016.2530041 |
[29] | X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63–69. https://doi.org/10.1016/j.automatica.2015.10.002 doi: 10.1016/j.automatica.2015.10.002 |
[30] | G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum, New York, 1987. |
[31] | O. A. Ladyzenskaya, N. N. Uraltreva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. |
[32] | A. C. Lazer, P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proceed. Am. Math. Soc., 111 (1991), 721–730. |
[33] | J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proceed. Royal Soc. Edinburgh, 128 (1998), 1389–1401. https://doi.org/10.1017/S0308210500027384 doi: 10.1017/S0308210500027384 |
[34] | K. Di, B. Yan, The existence of positive solution for singular Kirchhoff equation with two parameters, Boundary Value Probl., 40 (2019), 1–13. https://doi.org/10.1186/s13661-019-1154-8 doi: 10.1186/s13661-019-1154-8 |
[35] | J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 36 (1977), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3 doi: 10.1016/S0304-0208(08)70870-3 |