Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $

Xing Yi; Shuhou Ye; Xing Yi; Shuhou Ye

doi:10.3934/era.2023269

Electronic Research Archive

2023, Volume 31, Issue 9: 5286-5312. doi: 10.3934/era.2023269

Previous Article Next Article

Theory article

Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $

Xing Yi ,
Shuhou Ye ^,

College of Mathematics and Statistics, Hunan Normal University, Hunan, China

Academic Editor: Alain Miranville

Received: 25 April 2023 Revised: 16 June 2023 Accepted: 07 July 2023 Published: 24 July 2023

In this paper, we study the following Kirchhoff-type system:

$ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $

where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.
- positive solutions,
- Kirchhoff-type systems,
- critical Sobolev exponent,
- concentration compactness principle,
- mountain pass lemma
Citation: Xing Yi, Shuhou Ye. Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in $ \mathbb{R}^3 $[J]. Electronic Research Archive, 2023, 31(9): 5286-5312. doi: 10.3934/era.2023269

Related Papers:

Abstract

In this paper, we study the following Kirchhoff-type system:

$ \begin{equation} \left\{ \begin{array}{ll} -(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $

where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.

References

[1]	T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573–598.
[2]	G. Rosen, Minimum value for $c$ in the Sobolev inequality $\left\\|\phi^3\right\\| \leq c\\|\nabla \phi\\|^3$, SIAM J. Appl. Math., 21 (1971), 30–32. https://doi.org/10.1137/0121004 doi: 10.1137/0121004
[3]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353–372. https://doi.org/10.1007/BF02418013 doi: 10.1007/BF02418013
[4]	G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincarè Anal., Non Lineairè, 9 (1992), 281–304. https://doi.org/10.1016/S0294-1449(16)30238-4
[5]	J. Liu, J. F. Liao, C. L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^{N }$, J. Math. Anal. Appl., 429 (2015), 1153–1172. https://doi.org/10.1016/j.jmaa.2015.04.066 doi: 10.1016/j.jmaa.2015.04.066
[6]	M. Willem, Minimax Theorems, Birkhäuser Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
[7]	P. G. Han, Mutiple positive solutions of nonhomogeneous elliptic systems involving critical Sobolev exponents, Nonlinear Anal., 64 (2006), 869–886. https://doi.org/10.1016/j.na.2005.04.053 doi: 10.1016/j.na.2005.04.053
[8]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[9]	M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
[10]	S. Bernstein, Sur une class dẑquations fonctionnelles aux dẑrivẑes partielles, Izv. Akad. Nauk SSSR Ser. Mat., 4 (1940), 17–26.
[11]	S. Pohoẑaev, A certain class of quasilinear hyperbolic equations, Mat. Sb., 96 (1975), 152–166.
[12]	J. Lions, On some questions in boundary value problems of mathematical physics, in North-Holland Mathematics Studies North-Holland (eds. G. M. De La Penha, L. A. J. Medeiros), North-Holland, 30 (1978), 284–346. https://doi.org/10.1016/S0304-0208(08)70870-3
[13]	C. O. Alves, F. J. S. A. Corrêa, G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equation Appl., 3 (2010), 409-417. https://doi.org/10.7153/dea-02-25 doi: 10.7153/dea-02-25
[14]	Q. Y. Dai, E. H. Lan, F. L. Shi, A priori bounds for positive solutions of Kirchhoff type equations, Comput. Math. Appl., 76 (2018), 1525–1534. https://doi.org/10.1016/j.camwa.2018.07.004 doi: 10.1016/j.camwa.2018.07.004
[15]	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. https://doi.org/10.1016/j.jmaa.2012.12.053 doi: 10.1016/j.jmaa.2012.12.053
[16]	A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156–170. https://doi.org/10.1016/j.na.2013.08.011 doi: 10.1016/j.na.2013.08.011
[17]	S. Gupta, G. Dwivedi, Ground state solution to N-Kirchhoff equation with critical exponential growth and without Ambrosetti-Rabinowitz condition, Rend. Circ. Mat. Palermo II. Ser., 2023 (2023), 1–12. https://doi.org/10.1007/s12215-023-00902-7 doi: 10.1007/s12215-023-00902-7
[18]	W. He, D. Qin, Q. Wu, Existence, multiplicity and nonexistence results for Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 616–635. https://doi.org/10.1515/anona-2020-0154 doi: 10.1515/anona-2020-0154
[19]	S. H. Liang, S. Y. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in $\mathbb{R}^N$, Nonlinear Anal., 81 (2013), 31-41. https://doi.org/10.1016/j.na.2012.12.003 doi: 10.1016/j.na.2012.12.003
[20]	S. H. Liang, J. H. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 17 (2014), 126–136. https://doi.org/10.1016/j.nonrwa.2013.10.011 doi: 10.1016/j.nonrwa.2013.10.011
[21]	P. Lv, G. Lin, X. Lv, The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping, Dem. Math., 56 (2023), 20220197. https://doi.org/10.1515/dema-2022-0197 doi: 10.1515/dema-2022-0197
[22]	T. Mukherjee, P. Pucci, M. Xiang, Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems, Discrete Cont. Dyn. Syst., 2022 (2022), 163–187. https://doi.org/10.3934/dcds.2021111 doi: 10.3934/dcds.2021111
[23]	D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differ. Equations, 257 (2014), 168–1193. https://doi.org/10.1016/j.jde.2014.05.002 doi: 10.1016/j.jde.2014.05.002
[24]	E. Toscano, C. Vetro, D. Wardowski, Systems of Kirchhoff type equations with gradient dependence in the reaction term via subsolution-supersolution method, Discrete Cont. Dyn. Syst. S, 2023 (2023). https://doi.org/10.3934/dcdss.2023070
[25]	X. H. Tang, S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var., 110 (2017). https://doi.org/10.1007/s00526-017-1214-9
[26]	C. O. Alves, D. C. de Morais Filho, M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., 42 (2000), 771–787. https://doi.org/10.1016/S0362-546X(99)00121-2 doi: 10.1016/S0362-546X(99)00121-2
[27]	J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, (1989).
[28]	B. Cheng, X. Wu, J. Liu, Multiple solutions for a class of Kirchhoff type problem with concave nonlinearity, Nonlinear Differ. Equation Appl., 19 (2012), 521–537. https://doi.org/10.1007/s00030-011-0141-2 doi: 10.1007/s00030-011-0141-2
[29]	J. Chabrowski, J. Yang, On the Neumana problem for an elliptic system of equations involving the critical Sobolev exponent, Colloq. Math., 90 (2001), 19–35. https://doi.org/10.4064/cm90-1-2 doi: 10.4064/cm90-1-2

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)