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doi: 10.3934/dcds.2022071
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## On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

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

*Corresponding author: Shengbing Deng

Received  January 2022 Revised  May 2022 Early access May 2022

Fund Project: The authors have been supported by NSFC 11971392, Natural Science Foundation of Chongqing, China cstc2019jcyjjqX0022 and Fundamental Research Funds for the Central Universities XDJK2019TY001

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems
 $\begin{eqnarray*} \left\{ \begin{array}{ll} -m\left(\mathop \sum \limits_{j = 1}^k \|u_j\|^2\right)\Delta u_i = \frac{f_i(x, u_1, \ldots, u_k)}{|x|^\beta}+\varepsilon h_i(x), \ \ & \mbox{in}\ \ \Omega, \ \ i = 1, \ldots, k , \\ u_1 = u_2 = \cdots = u_k = 0, \ \ & \mbox{on}\ \ \partial\Omega, \end{array} \right. \end{eqnarray*}$
where
 $\Omega$
is a bounded domain in
 $\mathbb{R}^2$
containing the origin with smooth boundary,
 $\beta\in [0, 2)$
,
 $m$
is a Kirchhoff type function,
 $\|u_j\|^2 = \int_\Omega|\nabla u_j|^2dx$
,
 $f_i$
behaves like
 $e^{\alpha_0 s^2}$
when
 $|s|\rightarrow \infty$
for some
 $\alpha_0>0$
, and there is
 $C^1$
function
 $F: \Omega\times\mathbb{R}^k\to \mathbb{R}$
such that
 $\left(\frac{\partial F}{\partial u_1}, \ldots, \frac{\partial F}{\partial u_k}\right) = \left(f_1, \ldots, f_k\right)$
,
 $h_i\in \left(\big(H^1_0(\Omega)\big)^*, \|\cdot\|_*\right)$
. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when
 $\varepsilon>0$
is small.
Citation: Shengbing Deng, Xingliang Tian. On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022071
##### References:
 [1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $N$-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1990), 393-413. [2] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9. [3] Adimurthi, P. N. Skikanth and S. L. Yadava, Phenomena of critical exponent in ${\mathbb {R}}^2$, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 19-25.  doi: 10.1017/S0308210500028274. [4] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of ${\mathbb {R}}^2$ involving critical exponent, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1990), 481-504. [5] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, (2010), 2394-2426.  doi: 10.1093/imrn/rnp194. [6] F. S. B Albuquerque and E. S. Medeiros, An elliptic equation involving exponential critical growth in $\mathbb{R}^2$, Adv. Nonlinear Stud., 15 (2015), 23-37.  doi: 10.1515/ans-2015-0102. [7] 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. [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of ground state, Arch. Ration Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [9] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [10] W. Chen, Existence of solutions for fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity, Topol. Methods Nonlinear Anal., 54 (2019), 773-791.  doi: 10.12775/tmna.2019.069. [11] W. Chen and F. Yu, On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two, Appl. Anal., 101 (2022), 421-436.  doi: 10.1080/00036811.2020.1745778. [12] B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.  doi: 10.1016/j.jmaa.2012.04.025. [13] D. G de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math., 55 (2002), 135-152.  doi: 10.1002/cpa.10015. [14] D. G de Figueiredo, J. M. do Ó and B. Ruf, Semilinear elliptic systems with exponential nonlinearities in two dimensions, Adv. Nonlinear Stud., 6 (2006), 199-213.  doi: 10.1515/ans-2006-0205. [15] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in ${\mathbb {R}}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003. [16] M. de Souza, On a singular class of elliptic systems involving critical growth in $\mathbb{R}^2$, Nonlinear Anal. Real World Appl., 12 (2011), 1072-1088.  doi: 10.1016/j.nonrwa.2010.09.001. [17] J. M. do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in ${\mathbb {R}}^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979. [18] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^N$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020. [19] J. M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074. [20] G. M. Figueiredo and U. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math., 84 (2016), 23-39.  doi: 10.1007/s00032-015-0248-8. [21] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [22] X. He and W. Zou, Infinitely many positive solutions of Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.  doi: 10.1016/j.na.2008.02.021. [23] X.-M. He and W.-M. Zou, Multiplicity solutions of for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394.  doi: 10.1007/s10255-010-0005-2. [24] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [25] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [26] J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [27] D. Naimen and C. Tarsi, Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth, Adv. Differential Equations, 22 (2017), 983-1012. [28] P. Rabelo, Elliptic Systems involving critical growth in dimension two, Commun. Pure Appl. Anal., 8 (2009), 2013-2035.  doi: 10.3934/cpaa.2009.8.2013. [29] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028. [30] Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704.  doi: 10.1016/j.jfa.2011.11.018.

show all references

##### References:
 [1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $N$-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1990), 393-413. [2] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9. [3] Adimurthi, P. N. Skikanth and S. L. Yadava, Phenomena of critical exponent in ${\mathbb {R}}^2$, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 19-25.  doi: 10.1017/S0308210500028274. [4] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of ${\mathbb {R}}^2$ involving critical exponent, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1990), 481-504. [5] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, (2010), 2394-2426.  doi: 10.1093/imrn/rnp194. [6] F. S. B Albuquerque and E. S. Medeiros, An elliptic equation involving exponential critical growth in $\mathbb{R}^2$, Adv. Nonlinear Stud., 15 (2015), 23-37.  doi: 10.1515/ans-2015-0102. [7] 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. [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of ground state, Arch. Ration Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [9] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [10] W. Chen, Existence of solutions for fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity, Topol. Methods Nonlinear Anal., 54 (2019), 773-791.  doi: 10.12775/tmna.2019.069. [11] W. Chen and F. Yu, On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two, Appl. Anal., 101 (2022), 421-436.  doi: 10.1080/00036811.2020.1745778. [12] B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.  doi: 10.1016/j.jmaa.2012.04.025. [13] D. G de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math., 55 (2002), 135-152.  doi: 10.1002/cpa.10015. [14] D. G de Figueiredo, J. M. do Ó and B. Ruf, Semilinear elliptic systems with exponential nonlinearities in two dimensions, Adv. Nonlinear Stud., 6 (2006), 199-213.  doi: 10.1515/ans-2006-0205. [15] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in ${\mathbb {R}}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003. [16] M. de Souza, On a singular class of elliptic systems involving critical growth in $\mathbb{R}^2$, Nonlinear Anal. Real World Appl., 12 (2011), 1072-1088.  doi: 10.1016/j.nonrwa.2010.09.001. [17] J. M. do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in ${\mathbb {R}}^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979. [18] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^N$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020. [19] J. M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074. [20] G. M. Figueiredo and U. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math., 84 (2016), 23-39.  doi: 10.1007/s00032-015-0248-8. [21] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [22] X. He and W. Zou, Infinitely many positive solutions of Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414.  doi: 10.1016/j.na.2008.02.021. [23] X.-M. He and W.-M. Zou, Multiplicity solutions of for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394.  doi: 10.1007/s10255-010-0005-2. [24] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [25] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [26] J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J., 20 (1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [27] D. Naimen and C. Tarsi, Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth, Adv. Differential Equations, 22 (2017), 983-1012. [28] P. Rabelo, Elliptic Systems involving critical growth in dimension two, Commun. Pure Appl. Anal., 8 (2009), 2013-2035.  doi: 10.3934/cpaa.2009.8.2013. [29] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028. [30] Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704.  doi: 10.1016/j.jfa.2011.11.018.
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