doi: 10.3934/cpaa.2022082
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Ground state solutions for asymptotically periodic nonlinearities for Kirchhoff problems

Universidade Federal de Gois, 74001-970, Gois-GO, Brazil

* Corresponding author

Received  November 2021 Revised  February 2022 Early access April 2022

Fund Project: The first author was partially supported by CNPq with grants 309026/2020-2 and 429955/2018-9

It is establish existence of ground state solutions for nonlocal elliptic problems driven by Kirchhoff problem in the following form:
$ \begin{equation*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u +V(x)u = \lambda q(x)u + g(x, u), \; \; \; x \in \mathbb{R}^3, u \in H^{1}(\mathbb{R}^{3}). \end{equation*} $
where the potential
$ V $
and nonlinearity
$ g $
are periodic or asymptotically periodic. The main difficulty is to handle the lack of compactness due to the invariance under translations. The approach is based on minimization arguments over the Nehari set taking into account the fibering maps. Furthermore, due to the lack of compactness of Sobolev embedding into Lebesgue spaces, we need to recovery some kind of compactness required in variational methods. In order to do that we apply some fine estimates together with Lions' Lemma.
Citation: Edcarlos D. Silva, Jefferson S. Silva. Ground state solutions for asymptotically periodic nonlinearities for Kirchhoff problems. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022082
References:
[1]

C. O. AlvesF. J. S. A. Corr$\hat{e}$a 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.

[2]

A. Ambrosetti and D. Arcoya, Positive solutions of elliptic Kirchhoff equations, Adv. Nonlinear Stud., 17 (2017), 3-15.  doi: 10.1515/ans-2016-6004.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725–1741. doi: 10.1080/03605309508821149.

[4]

C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876-1908.  doi: 10.1016/j.jde.2010.11.017.

[5]

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.

[6]

B. ChengX. Wu and J. Liu, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, Nonlinear Differ. Equ. Appl., 19 (2012), 521-537.  doi: 10.1007/s00030-011-0141-2.

[7]

G. Eisley, Nonlinear vibrations of beams and rectangular plates., Z. Anger. Math. Phys., 15 (1964), 167-175.  doi: 10.1007/BF01602658.

[8]

G. M. Figueiredo and M. T. O. Pimenta, Existence of ground state solutions to Dirac equations with Vanishing potential at infinity., J. Differ. Equ., 262 (2017), 486-505.  doi: 10.1016/j.jde.2016.09.034.

[9]

M. F. Furtado, L. D. Oliveira and J. P. P. da Silva, Multiple solutions for a Kirchhoff equation with critical growth, Z. Angew. Math. Phys., 70 (2019), 15 pp. doi: 10.1007/s00033-018-1045-3.

[10]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 2 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.

[11]

J. Limaco and L. A. Medeiros, Kirchhoff-Carrier elastic strings in noncylindrical domains, Portug. Math., 14 (1999), 464-500. 

[12]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[13]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, , Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, (1984), 109–283.

[14]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.  doi: 10.1016/S0304-0208(08)70870-3.

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[16]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. 

[17]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[18]

E. D. Silva and J. S. Silva, Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues, J. Math. Phys., 60 (2019), 24pp. doi: 10.1063/1.5091810.

[19]

E. D. Silva and J. S. Silva, Existence of solution for quasilinear Schrödinger equations using a linking structure, Complex Variables and Elliptic Equations, 67 (2022), 401–432. doi: 10.1080/17476933.2020.1833867.

[20]

E. D. Silva and J. S. Silva, Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure, Discrete Contin. Dyn. Syst., 40 (2020), 5441-5470.  doi: 10.3934/dcds.2020234.

[21]

J. Sun and T. Wu., Ground state solutions for an indefinite Kirchhoff type problem with steep potential well., J. Differ. Equ., 256 (2014), 1771-1792.  doi: 10.1016/j.jde.2013.12.006.

[22]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[23]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, 2010.

[24]

J. WangL. X. TianJ. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023.

[25]

M. Willem, Minimax Theorems, Birkhauser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.

show all references

References:
[1]

C. O. AlvesF. J. S. A. Corr$\hat{e}$a 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.

[2]

A. Ambrosetti and D. Arcoya, Positive solutions of elliptic Kirchhoff equations, Adv. Nonlinear Stud., 17 (2017), 3-15.  doi: 10.1515/ans-2016-6004.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725–1741. doi: 10.1080/03605309508821149.

[4]

C. ChenY. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250 (2011), 1876-1908.  doi: 10.1016/j.jde.2010.11.017.

[5]

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.

[6]

B. ChengX. Wu and J. Liu, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, Nonlinear Differ. Equ. Appl., 19 (2012), 521-537.  doi: 10.1007/s00030-011-0141-2.

[7]

G. Eisley, Nonlinear vibrations of beams and rectangular plates., Z. Anger. Math. Phys., 15 (1964), 167-175.  doi: 10.1007/BF01602658.

[8]

G. M. Figueiredo and M. T. O. Pimenta, Existence of ground state solutions to Dirac equations with Vanishing potential at infinity., J. Differ. Equ., 262 (2017), 486-505.  doi: 10.1016/j.jde.2016.09.034.

[9]

M. F. Furtado, L. D. Oliveira and J. P. P. da Silva, Multiple solutions for a Kirchhoff equation with critical growth, Z. Angew. Math. Phys., 70 (2019), 15 pp. doi: 10.1007/s00033-018-1045-3.

[10]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 2 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.

[11]

J. Limaco and L. A. Medeiros, Kirchhoff-Carrier elastic strings in noncylindrical domains, Portug. Math., 14 (1999), 464-500. 

[12]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[13]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, , Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, (1984), 109–283.

[14]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.  doi: 10.1016/S0304-0208(08)70870-3.

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[16]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. 

[17]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[18]

E. D. Silva and J. S. Silva, Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues, J. Math. Phys., 60 (2019), 24pp. doi: 10.1063/1.5091810.

[19]

E. D. Silva and J. S. Silva, Existence of solution for quasilinear Schrödinger equations using a linking structure, Complex Variables and Elliptic Equations, 67 (2022), 401–432. doi: 10.1080/17476933.2020.1833867.

[20]

E. D. Silva and J. S. Silva, Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure, Discrete Contin. Dyn. Syst., 40 (2020), 5441-5470.  doi: 10.3934/dcds.2020234.

[21]

J. Sun and T. Wu., Ground state solutions for an indefinite Kirchhoff type problem with steep potential well., J. Differ. Equ., 256 (2014), 1771-1792.  doi: 10.1016/j.jde.2013.12.006.

[22]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[23]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, 2010.

[24]

J. WangL. X. TianJ. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023.

[25]

M. Willem, Minimax Theorems, Birkhauser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[26]

Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.

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