A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in \mathbb{R}^2

Previous Article
Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids
CPAA Home
This Issue
Next Article
On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol

July 2016, 15(4): 1351-1370. doi: 10.3934/cpaa.2016.15.1351

A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $

1.	Institut Supérieur des Mathématiques Appliquées et de l'Informatique de Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisia

Received November 2015 Revised January 2016 Published April 2016

Abstract
Related Papers

In this paper, we consider the existence and multiplicity of sign-changing solutions to some Kirchhoff-type equation involving a nonlinear term with exponential growth. In a first result, we prove the existence of at least three solutions: one solution is positive, one is negative and the third one is sign-changing. The existence of infinitely many sign-changing solutions is proved as our second result in this work. Our method is mainly based on invariant sets of descending flow in the framework of classical critical point theory.

Keywords: multiplicity, Kirchhoff-type, sign-changing solution, invariant sets, descending flow, critical points., Trudinger-Moser inequality.

Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35A15, 35B08, 35D3.

Citation: Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1351-1370. doi: 10.3934/cpaa.2016.15.1351

References:

[1]	Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbbR^2$ involving critical exponent,, \emph{Ann. Scuola. Norm. Sup. Pisa}, 17 (1990), 481. Google Scholar
[2]	C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $ \mathbbR^2 $ with Neumann conditions,, \emph{J. Differential Equations}, 219 (2005), 20. doi: 10.1016/j.jde.2004.11.010. Google Scholar
[3]	C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $ \mathbbR^2, $, \emph{Topol. Methods Nonlinear Anal.}, 46 (2015), 867. Google Scholar
[4]	C.O. Alves and D.S. Pereira, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in $\mathbbR^2,$, (2014), (2014). Google Scholar
[5]	C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth., \emph{J. Differential Equation}, 234 (2007), 464. Google Scholar
[6]	S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth,, \emph{Electon. J. Differential Equations}, 2014 (2014), 1. Google Scholar
[7]	S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $ \mathbbR^N, $, \emph{Nonlinear Stud.}, 22 (2015), 57. Google Scholar
[8]	S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition,, \emph{Nonlinear Anal.}, 125 (2015), 626. Google Scholar
[9]	T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1. Google Scholar
[10]	T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 22 (2005), 259. doi: 10.1016/j.anihpc.2004.07.005. Google Scholar
[11]	T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 13 (1999), 191. Google Scholar
[12]	T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, \emph{Topol. Methods Nonlinear Anal.}, 22 (2003), 1. Google Scholar
[13]	C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, (2015) arXiv:1501.05733v1., (2015). Google Scholar
[14]	C.J. Batkam, An elliptic equation under the effect of two nonlocal terms,, \emph{Math. Meth. Appl. Sci.}, (2015). Google Scholar
[15]	H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313. Google Scholar
[16]	H. Brezis, Analyse Fonctionnelle (théorie et applications),, Masson, (1983). Google Scholar
[17]	D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR^2 $,, \emph{Commun. Partial Differ. Equ.}, 17 (1992), 407. Google Scholar
[18]	E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity,, \emph{J. Math. Anal. Appl.}, 250 (2000), 449. doi: 10.1006/jmaa.2000.6969. Google Scholar
[19]	J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth,, \emph{Abstr. Appl. Anal.}, 2 (1997), 301. doi: 10.1155/S1085337597000419. Google Scholar
[20]	J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,, \emph{J. Math. Anal. Appl.}, 345 (2008), 286. doi: 10.1016/j.jmaa.2008.03.074. Google Scholar
[21]	G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation,, \emph{Math. Nachr.}, 288 (2015), 48. Google Scholar
[22]	G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth,, \emph{Milan J. Math.}, (2015). doi: 10.10007/s00032-015-0248-8. Google Scholar
[23]	M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883. doi: 10.1016/j.jde.2008.07.004. Google Scholar
[24]	Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 117 (2015), 159. doi: 10.1016/j.na.2015.01.005. Google Scholar
[25]	X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations,, \emph{Discrete Dyn. Nat. Soc.}, (2015), 1. Google Scholar
[26]	Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257. Google Scholar
[27]	A. Mao and S. Yuan, 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. Google Scholar
[28]	D. Mugnai, Four nontrivial solutions for subcritical exponential equation,, \emph{Calc. Var. Partial Differential Equations}, 32 (2008), 480. doi: 10.1007/s00526-007-0148-z. Google Scholar
[29]	R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems,, \emph{Abstr. Appl. Anal}, 2014 (2014). doi: 10.1155/2014/163645. Google Scholar
[30]	K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities,, \emph{Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM}, (2015), 13398. Google Scholar
[31]	S. Struwe, Variational Methods,, Springer-Verlag, (2000). doi: 10.1007/978-3-662-04194-9. Google Scholar
[32]	N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, XX (1967), 721. Google Scholar
[33]	T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones,, \emph{Topol. Methods Nonlinear Anal.}, 28 (2006), 33. Google Scholar
[34]	Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities,, \emph{J. Math. Anal. Appl.}, 401 (2013), 850. doi: 10.1016/j.jmaa.2013.01.006. Google Scholar
[35]	W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 427 (2015), 722. doi: 10.1016/j.jmaa.2015.02.070. Google Scholar
[36]	Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth,, \emph{Commun. Contemp. Math.}, 6 (2004), 947. doi: 10.1142/S0219199704001549. Google Scholar
[37]	Z. 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. Google Scholar

show all references

References:

[1]	Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbbR^2$ involving critical exponent,, \emph{Ann. Scuola. Norm. Sup. Pisa}, 17 (1990), 481. Google Scholar
[2]	C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $ \mathbbR^2 $ with Neumann conditions,, \emph{J. Differential Equations}, 219 (2005), 20. doi: 10.1016/j.jde.2004.11.010. Google Scholar
[3]	C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $ \mathbbR^2, $, \emph{Topol. Methods Nonlinear Anal.}, 46 (2015), 867. Google Scholar
[4]	C.O. Alves and D.S. Pereira, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in $\mathbbR^2,$, (2014), (2014). Google Scholar
[5]	C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth., \emph{J. Differential Equation}, 234 (2007), 464. Google Scholar
[6]	S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth,, \emph{Electon. J. Differential Equations}, 2014 (2014), 1. Google Scholar
[7]	S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $ \mathbbR^N, $, \emph{Nonlinear Stud.}, 22 (2015), 57. Google Scholar
[8]	S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition,, \emph{Nonlinear Anal.}, 125 (2015), 626. Google Scholar
[9]	T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1. Google Scholar
[10]	T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 22 (2005), 259. doi: 10.1016/j.anihpc.2004.07.005. Google Scholar
[11]	T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 13 (1999), 191. Google Scholar
[12]	T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, \emph{Topol. Methods Nonlinear Anal.}, 22 (2003), 1. Google Scholar
[13]	C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, (2015) arXiv:1501.05733v1., (2015). Google Scholar
[14]	C.J. Batkam, An elliptic equation under the effect of two nonlocal terms,, \emph{Math. Meth. Appl. Sci.}, (2015). Google Scholar
[15]	H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313. Google Scholar
[16]	H. Brezis, Analyse Fonctionnelle (théorie et applications),, Masson, (1983). Google Scholar
[17]	D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR^2 $,, \emph{Commun. Partial Differ. Equ.}, 17 (1992), 407. Google Scholar
[18]	E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity,, \emph{J. Math. Anal. Appl.}, 250 (2000), 449. doi: 10.1006/jmaa.2000.6969. Google Scholar
[19]	J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth,, \emph{Abstr. Appl. Anal.}, 2 (1997), 301. doi: 10.1155/S1085337597000419. Google Scholar
[20]	J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,, \emph{J. Math. Anal. Appl.}, 345 (2008), 286. doi: 10.1016/j.jmaa.2008.03.074. Google Scholar
[21]	G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation,, \emph{Math. Nachr.}, 288 (2015), 48. Google Scholar
[22]	G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth,, \emph{Milan J. Math.}, (2015). doi: 10.10007/s00032-015-0248-8. Google Scholar
[23]	M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883. doi: 10.1016/j.jde.2008.07.004. Google Scholar
[24]	Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 117 (2015), 159. doi: 10.1016/j.na.2015.01.005. Google Scholar
[25]	X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations,, \emph{Discrete Dyn. Nat. Soc.}, (2015), 1. Google Scholar
[26]	Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257. Google Scholar
[27]	A. Mao and S. Yuan, 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. Google Scholar
[28]	D. Mugnai, Four nontrivial solutions for subcritical exponential equation,, \emph{Calc. Var. Partial Differential Equations}, 32 (2008), 480. doi: 10.1007/s00526-007-0148-z. Google Scholar
[29]	R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems,, \emph{Abstr. Appl. Anal}, 2014 (2014). doi: 10.1155/2014/163645. Google Scholar
[30]	K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities,, \emph{Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM}, (2015), 13398. Google Scholar
[31]	S. Struwe, Variational Methods,, Springer-Verlag, (2000). doi: 10.1007/978-3-662-04194-9. Google Scholar
[32]	N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, XX (1967), 721. Google Scholar
[33]	T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones,, \emph{Topol. Methods Nonlinear Anal.}, 28 (2006), 33. Google Scholar
[34]	Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities,, \emph{J. Math. Anal. Appl.}, 401 (2013), 850. doi: 10.1016/j.jmaa.2013.01.006. Google Scholar
[35]	W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 427 (2015), 722. doi: 10.1016/j.jmaa.2015.02.070. Google Scholar
[36]	Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth,, \emph{Commun. Contemp. Math.}, 6 (2004), 947. doi: 10.1142/S0219199704001549. Google Scholar
[37]	Z. 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. Google Scholar

[1]	Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032
[2]	Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[3]	Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137
[4]	Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378
[5]	Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[6]	Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
[7]	Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
[8]	Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883
[9]	Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256
[10]	Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499
[11]	Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124
[12]	Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445
[13]	Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031
[14]	Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[15]	Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883
[16]	Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[17]	Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[18]	Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[19]	Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[20]	Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences