# American Institute of Mathematical Sciences

September  2016, 15(5): 1841-1856. doi: 10.3934/cpaa.2016006

## Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China

Received  August 2015 Revised  April 2016 Published  July 2016

In this article, we study the existence and multiplicity of positive solutions for the Kirchhoff type problem with singular and critical nonlinearities \begin{eqnarray} \begin{cases} -\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{eqnarray} where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.
Citation: Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
##### References:
 [1] 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., 23 (2010), 409-417. doi: 10.7153/dea-02-25. [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), 85-93. doi: 10.1016/j.camwa.2005.01.008. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. [4] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [5] B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Annal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025. [6] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065. [7] 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), 706-713. doi: 10.1016/j.jmaa.2012.12.053. [8] A. Hamydy, M. Massar and N. Tsouli, Existence of solution for $p$-Kirchhoff type problems with critical exponents, Electronic J. Differential Equations, 105 (2011), 1-8. [9] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Annal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [10] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [11] C. Y. Lei, J. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538. doi: 10.1016/j.jmaa.2014.07.031. [12] J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbbR^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172. doi: 10.1016/j.jmaa.2015.04.066. [13] J. F. Liao, P. Zhang, J. Liu and C. L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148. doi: 10.1016/j.jmaa.2015.05.038. [14] J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat, Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud. vol. 30, North-Holland, Amsterdam, 1978, pp. 284-346. [15] S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbbR^N$, Nonlinear Anal., 81 (2013), 31-41. doi: 10.1016/j.na.2012.12.003. [16] S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical growth in $\mathbbR^3$, Nonlinear Anal. Real World Appl., 17 (2014), 126-136. doi: 10.1016/j.nonrwa.2013.10.011. [17] X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733. [18] Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [19] A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021. [20] A. M. Mao and Z. T. 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. [21] T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1. [22] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193. doi: 10.1016/j.jde.2014.05.002. [23] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004. [24] J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061. [25] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. [26] L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1. [27] 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), 2773-2786. doi: 10.3934/cpaa.2013.12.2773. [28] Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30. doi: 10.1016/j.na.2010.02.008. [29] Z. T. Zhang and 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.

show all references

##### References:
 [1] 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., 23 (2010), 409-417. doi: 10.7153/dea-02-25. [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), 85-93. doi: 10.1016/j.camwa.2005.01.008. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. [4] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [5] B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Annal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025. [6] B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065. [7] 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), 706-713. doi: 10.1016/j.jmaa.2012.12.053. [8] A. Hamydy, M. Massar and N. Tsouli, Existence of solution for $p$-Kirchhoff type problems with critical exponents, Electronic J. Differential Equations, 105 (2011), 1-8. [9] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Annal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [10] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [11] C. Y. Lei, J. F. Liao and C. L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538. doi: 10.1016/j.jmaa.2014.07.031. [12] J. Liu, J. F. Liao and C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbbR^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172. doi: 10.1016/j.jmaa.2015.04.066. [13] J. F. Liao, P. Zhang, J. Liu and C. L. Tang, Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity, J. Math. Anal. Appl., 430 (2015), 1124-1148. doi: 10.1016/j.jmaa.2015.05.038. [14] J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat, Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud. vol. 30, North-Holland, Amsterdam, 1978, pp. 284-346. [15] S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbbR^N$, Nonlinear Anal., 81 (2013), 31-41. doi: 10.1016/j.na.2012.12.003. [16] S. H. Liang and J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical growth in $\mathbbR^3$, Nonlinear Anal. Real World Appl., 17 (2014), 126-136. doi: 10.1016/j.nonrwa.2013.10.011. [17] X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733. [18] Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [19] A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021. [20] A. M. Mao and Z. T. 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. [21] T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1. [22] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193. doi: 10.1016/j.jde.2014.05.002. [23] J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004. [24] J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061. [25] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. [26] L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1. [27] 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), 2773-2786. doi: 10.3934/cpaa.2013.12.2773. [28] Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30. doi: 10.1016/j.na.2010.02.008. [29] Z. T. Zhang and 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.
 [1] Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773 [2] Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721 [3] Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007 [4] Qilin Xie, Jianshe Yu. Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension. Communications on Pure and Applied Analysis, 2019, 18 (1) : 129-158. doi: 10.3934/cpaa.2019008 [5] Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure and Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445 [6] Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure and Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567 [7] Mingqi Xiang, Binlin Zhang. A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 413-433. doi: 10.3934/dcdss.2019027 [8] Juncheng Wei, Ke Wu. Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022044 [9] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [10] Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143 [11] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [12] Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015 [13] Sondes khabthani, Lassaad Elasmi, François Feuillebois. Perturbation solution of the coupled Stokes-Darcy problem. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 971-990. doi: 10.3934/dcdsb.2011.15.971 [14] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [15] Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 [16] Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 [17] Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111 [18] Wenjing Chen. Multiplicity of solutions for a fractional Kirchhoff type problem. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2009-2020. doi: 10.3934/cpaa.2015.14.2009 [19] Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124 [20] To Fu Ma. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Conference Publications, 2007, 2007 (Special) : 694-703. doi: 10.3934/proc.2007.2007.694

2020 Impact Factor: 1.916