A Kirchhoff type elliptic systems with exponential growth nonlinearities
DOI:
https://doi.org/10.12775/TMNA.2021.035Keywords
Kirchhoff type elliptic systems, exponential growth nonlinearity, mountain-pass theorem, Trudinger-Moser inequalityAbstract
In this paper we are interested in the existence of solutions for the following Kirchhoff type elliptic systems \begin{equation*} \begin{cases} \displaystyle -M\Bigg(\sum^m_{j=1}\|u_j\|^2\Bigg)\Delta u_i=f_i(x,u_1,\ldots,u_m) &\mbox{in } \Omega,\\%[2mm] u_1=\ldots=u_m=0 &\mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^2$, $M$ is a Kirchhoff type function, $\|u_i\|^2:=\int_\Omega|\nabla u_i|^2{d}x$, $f_i$ behaves like $\exp(\beta s^2)$ when $|s|\rightarrow \infty$ for some $\beta> 0$, $i=1,\ldots,m$. By variational methods with the Trudinger-Moser inequality, we obtain the existence of solutions for the above systems.References
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.
Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger–Moser inequality in RN and its applications, Int. Math. Res. Notices 13 (2010), 2394–2426.
F.S.B. Albuquerque, Standing wave solutions for a class of nonhomogeneous systems in dimension two, Complex Var. Elliptic Equ. 61 (2016), no. 8, 1157–1175.
S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition, Nonlinear Anal. 125 (2015), 626–638.
R. Arora, J. Giacomoni, T. Mukherjee and K. Sreenadh, n-Kirchhoff–Choquard equations with exponential nonlinearity, Nonlinear Anal. 186 (2019), 113–144.
R. Arora, J. Giacomoni, T. Mukherjee and K. Sreenadh, Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications, Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. 114 (2020), no. 3, 1–26.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.
J. Chabrowski, Variational methods for potential operator equations. With applications to nonlinear elliptic equations, de Gruyter Studies in Mathematics, vol. 24, Walter de Gruyter, Berlin, 1997.
W. Chen, Existence of solutions for fractional p-Kirchhoff type equations with a generalized Choquard nonlinearities, Topol. Methods Nonlinear Anal. 54 (2019), 773–791.
W. Chen and F. Yu, On a fractional Kirchhoff type problem with critical exponential growth nonlinearity, Appl. Math. Lett. 105 (2020), 106279.
W. Chen and F. Yu, On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two, Appl. Anal. (2020), 1-16.
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.
D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. 3 (1995), 139–153.
M. de Souza, On a singular class of elliptic systems involving critical growth in R2 , Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1072–1088.
M. de Souza, On a class of nonhomogeneous elliptic equation on compact Riemannian manifold without boundary, Mediterr. J. Math. 15(3) (2018), 101.
J.M. do Ó, Semilinear Dirichlet problems for the N -Laplacian in RN with nonlinearities in the critical growth range, Differ. Integral Equ. 9 (1996), 967–979. [17] J. M. do Ó, E. Medeiros, U.B. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), 286-304.
G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), 23–39.
P. Lions, The concentration-compactness principle in the calculus of variations, The limit case I, Rev. Mat. Iberoamericana 1 (1985), 145–201.
Q. Li and Z. Yang, Multiple solutions for N -Kirchhoff type problems with critical exponential growth in RN , Nonlinear Anal. 117 (2015), 159–168.
P.K. Mishra, S. Goyal and K. Sreenadh, Polyharmonic Kirchhoff type equations with singular exponential nonlinearities, Commun. Pure Appl. Anal. 15 (2016), no. 5, 1689–1717.
J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J. 20 (1971), 1077–1092.
X. Mingqi, V. D. Rǎdulescu and B. Zhang, Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity, Calc. Var. Partial Differential Equations 58) (2019), no. 2, 57.
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.
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65, Amer. Math. Soc., Providence, 1986.
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.
N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Xingliang Tian
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
