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 & 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,, \emph{Differ. Equ. Appl.}, 23 (2010), 409. 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,, \emph{Comput. Math. Appl.}, 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[5]

B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, \emph{J. Math. Annal. Appl.}, 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025.

[6]

B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, \emph{Nonlinear Anal.}, 71 (2009), 4883. 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,, \emph{J. Math. Anal. Appl.}, 401 (2013), 706. 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,, \emph{Electronic J. Differential Equations}, 105 (2011), 1.

[9]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, \emph{J. Math. Annal. Appl.}, 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059.

[10]

G. Kirchhoff, Mechanik,, Teubner, (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,, \emph{J. Math. Anal. Appl.}, 421 (2015), 521. 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 $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 429 (2015), 1153. 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,, \emph{J. Math. Anal. Appl.}, 430 (2015), 1124. doi: 10.1016/j.jmaa.2015.05.038.

[14]

J. L. Lions, On some questions in boundary value problems of mathematical physics,, in \emph{Contemporary Developments in Continuum Mechanics and Partial Differential Equations}, (1997), 284.

[15]

S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 81 (2013), 31. 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 $\mathbbR^3$,, \emph{Nonlinear Anal. Real World Appl.}, 17 (2014), 126. doi: 10.1016/j.nonrwa.2013.10.011.

[17]

X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 721.

[18]

Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285. 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,, \emph{J. Math. Anal. Appl.}, 383 (2011), 239. 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,, \emph{Nonlinear Anal.}, 70 (2009), 1275. 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,, \emph{Appl. Math. Lett.}, 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1.

[22]

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four,, \emph{J. Differential Equations}, 257 (2014), 1168. 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,, \emph{Nonlinear Anal.}, 75 (2012), 3470. 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,, \emph{Nonlinear Anal.}, 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061.

[25]

G. Talenti, Best constant in Sobolev inequality,, \emph{Ann. Mat. Pura Appl.}, 110 (1976), 353.

[26]

L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, \emph{J. Appl. Math. Comput.}, 39 (2012), 473. 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,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773.

[28]

Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems,, \emph{Nonlinear Anal.}, 73 (2010), 25. 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,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456. 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,, \emph{Differ. Equ. Appl.}, 23 (2010), 409. 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,, \emph{Comput. Math. Appl.}, 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008.

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405.

[5]

B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,, \emph{J. Math. Annal. Appl.}, 394 (2012), 488. doi: 10.1016/j.jmaa.2012.04.025.

[6]

B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems,, \emph{Nonlinear Anal.}, 71 (2009), 4883. 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,, \emph{J. Math. Anal. Appl.}, 401 (2013), 706. 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,, \emph{Electronic J. Differential Equations}, 105 (2011), 1.

[9]

J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, \emph{J. Math. Annal. Appl.}, 369 (2010), 564. doi: 10.1016/j.jmaa.2010.03.059.

[10]

G. Kirchhoff, Mechanik,, Teubner, (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,, \emph{J. Math. Anal. Appl.}, 421 (2015), 521. 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 $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 429 (2015), 1153. 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,, \emph{J. Math. Anal. Appl.}, 430 (2015), 1124. doi: 10.1016/j.jmaa.2015.05.038.

[14]

J. L. Lions, On some questions in boundary value problems of mathematical physics,, in \emph{Contemporary Developments in Continuum Mechanics and Partial Differential Equations}, (1997), 284.

[15]

S. H. Liang and S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 81 (2013), 31. 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 $\mathbbR^3$,, \emph{Nonlinear Anal. Real World Appl.}, 17 (2014), 126. doi: 10.1016/j.nonrwa.2013.10.011.

[17]

X. Liu and Y. J. Sun, Multiple positive solutions for Kirchhoff type problems with singularity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 721.

[18]

Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285. 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,, \emph{J. Math. Anal. Appl.}, 383 (2011), 239. 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,, \emph{Nonlinear Anal.}, 70 (2009), 1275. 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,, \emph{Appl. Math. Lett.}, 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1.

[22]

D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four,, \emph{J. Differential Equations}, 257 (2014), 1168. 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,, \emph{Nonlinear Anal.}, 75 (2012), 3470. 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,, \emph{Nonlinear Anal.}, 74 (2011), 1212. doi: 10.1016/j.na.2010.09.061.

[25]

G. Talenti, Best constant in Sobolev inequality,, \emph{Ann. Mat. Pura Appl.}, 110 (1976), 353.

[26]

L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, \emph{J. Appl. Math. Comput.}, 39 (2012), 473. 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,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2773. doi: 10.3934/cpaa.2013.12.2773.

[28]

Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems,, \emph{Nonlinear Anal.}, 73 (2010), 25. 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,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456. doi: 10.1016/j.jmaa.2005.06.102.

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