# American Institute of Mathematical Sciences

May  2016, 21(3): 883-908. doi: 10.3934/dcdsb.2016.21.883

## Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions

 1 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 2 Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2 3 School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024

Received  April 2015 Revised  December 2015 Published  January 2016

This paper deals with a $p$-Kirchhoff type problem involving sign-changing weight functions. It is shown that under certain conditions, by means of variational methods, the existence of multiple nontrivial nonnegative solutions for the problem with the subcritical exponent are obtained. Moreover, in the case of critical exponent, we establish the existence of the solutions and prove that the elliptic equation possesses at least one nontrivial nonnegative solution.
Citation: Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
##### References:
 [1] C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of the problem involving a nonlinear operator,, Comm. Appl. Nonlinear Anal., 8 (2001), 43. Google Scholar [2] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar [3] C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differ. Equ. Appl., 2 (2010), 409. doi: 10.7153/dea-02-25. Google Scholar [4] G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem,, J. Math. Anal. Appl., 373 (2011), 248. doi: 10.1016/j.jmaa.2010.07.019. Google Scholar [5] K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function,, J. Differential Equations, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar [6] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar [7] M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems,, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447. Google Scholar [8] C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions,, J. Differential Equations, 250 (2011), 1876. doi: 10.1016/j.jde.2010.11.017. Google Scholar [9] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065. Google Scholar [10] F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus,, Appl. Math. Lett., 22 (2009), 819. doi: 10.1016/j.aml.2008.06.042. Google Scholar [11] F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods,, Bull. Austral. Math. Soc., 74 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [12] P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703. doi: 10.1017/S0308210500023787. Google Scholar [13] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [14] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument,, J. Math. Anal. Appl., 401 (2013), 706. doi: 10.1016/j.jmaa.2012.12.053. Google Scholar [15] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [16] X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar [17] J. C. Huang, C. S. Chen and Z. H. Xiu, Existence and multiplicity results for a $p$-Kirchhoff equation with a concave-convex term,, Appl. Math. Lett., 26 (2013), 1070. doi: 10.1016/j.aml.2013.06.001. Google Scholar [18] A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for $p$-Kirchhoff type problems with critical exponent,, Electronic J. Differential Equations, 2011 (2011), 1. Google Scholar [19] G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar [20] J. L. Lions, On some questions in boundary value problems of mathmatical physics,, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, 30 (1978), 284. Google Scholar [21] P. L. Lions, The concentration-compactness principle in the calculus of variations,, the limit case Part I Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar [22] D. C. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem,, Nonlinear Anal., 72 (2010), 302. doi: 10.1016/j.na.2009.06.052. Google Scholar [23] D. C. Liu and P. H. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation,, Nonlinear Anal., 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar [24] T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar [25] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, Nonlinear Anal., 63 (2005), 1967. doi: 10.1016/j.na.2005.03.021. Google Scholar [26] A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity,, C. R. Acad. Sci. Paris, 352 (2014), 295. doi: 10.1016/j.crma.2014.01.015. Google Scholar [27] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar [28] Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent,, Commun. Pure Appl. Anal., 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773. Google Scholar

show all references

##### References:
 [1] C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of the problem involving a nonlinear operator,, Comm. Appl. Nonlinear Anal., 8 (2001), 43. Google Scholar [2] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, Comput. Math. Appl., 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar [3] C. O. Alves, F. J. S. A. Corrêa and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth,, Differ. Equ. Appl., 2 (2010), 409. doi: 10.7153/dea-02-25. Google Scholar [4] G. Anello, A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem,, J. Math. Anal. Appl., 373 (2011), 248. doi: 10.1016/j.jmaa.2010.07.019. Google Scholar [5] K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function,, J. Differential Equations, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar [6] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems,, Nonlinear Anal., 30 (1997), 4619. doi: 10.1016/S0362-546X(97)00169-7. Google Scholar [7] M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems,, Rairo Modélisation Math. Anal. Numér., 26 (1992), 447. Google Scholar [8] C. Y. Chen, Y. C. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions,, J. Differential Equations, 250 (2011), 1876. doi: 10.1016/j.jde.2010.11.017. Google Scholar [9] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, Nonlinear Anal., 71 (2009), 4883. doi: 10.1016/j.na.2009.03.065. Google Scholar [10] F. J. S. A. Corrêa and G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii's genus,, Appl. Math. Lett., 22 (2009), 819. doi: 10.1016/j.aml.2008.06.042. Google Scholar [11] F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods,, Bull. Austral. Math. Soc., 74 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [12] P. Drábek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703. doi: 10.1017/S0308210500023787. Google Scholar [13] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar [14] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument,, J. Math. Anal. Appl., 401 (2013), 706. doi: 10.1016/j.jmaa.2012.12.053. Google Scholar [15] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [16] X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems,, Nonlinear Anal., 70 (2009), 1407. doi: 10.1016/j.na.2008.02.021. Google Scholar [17] J. C. Huang, C. S. Chen and Z. H. Xiu, Existence and multiplicity results for a $p$-Kirchhoff equation with a concave-convex term,, Appl. Math. Lett., 26 (2013), 1070. doi: 10.1016/j.aml.2013.06.001. Google Scholar [18] A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for $p$-Kirchhoff type problems with critical exponent,, Electronic J. Differential Equations, 2011 (2011), 1. Google Scholar [19] G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar [20] J. L. Lions, On some questions in boundary value problems of mathmatical physics,, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, 30 (1978), 284. Google Scholar [21] P. L. Lions, The concentration-compactness principle in the calculus of variations,, the limit case Part I Rev. Mat. Iberoamericana, 1 (1985), 145. doi: 10.4171/RMI/6. Google Scholar [22] D. C. Liu, On a p-Kirchhoff equation via fountain theorem and dual fountain theorem,, Nonlinear Anal., 72 (2010), 302. doi: 10.1016/j.na.2009.06.052. Google Scholar [23] D. C. Liu and P. H. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation,, Nonlinear Anal., 75 (2012), 5032. doi: 10.1016/j.na.2012.04.018. Google Scholar [24] T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, Appl. Math. Lett., 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar [25] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, Nonlinear Anal., 63 (2005), 1967. doi: 10.1016/j.na.2005.03.021. Google Scholar [26] A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity,, C. R. Acad. Sci. Paris, 352 (2014), 295. doi: 10.1016/j.crma.2014.01.015. Google Scholar [27] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index,, J. Differential Equations, 221 (2006), 246. doi: 10.1016/j.jde.2005.03.006. Google Scholar [28] Q. L. Xie, X. P. Wu and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent,, Commun. Pure Appl. Anal., 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773. Google Scholar
 [1] Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032 [2] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [3] Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697 [4] Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137 [5] Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 [6] Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389 [7] Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383 [8] Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565 [9] Angela Pistoia, Tonia Ricciardi. Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5651-5692. doi: 10.3934/dcds.2017245 [10] Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [11] M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057 [12] Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151 [13] Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737 [14] Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256 [15] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [16] J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427 [17] Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439 [18] Jijiang Sun, Shiwang Ma. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2317-2330. doi: 10.3934/cpaa.2014.13.2317 [19] Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917 [20] Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004

2018 Impact Factor: 1.008