# American Institute of Mathematical Sciences

November 2018, 17(6): 2623-2638. doi: 10.3934/cpaa.2018124

## Ground states for Kirchhoff-type equations with critical growth

 1 Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China 2 Department of Mathematics, Tsinghua University, Beijing, Beijing 100084, China 3 Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China 4 Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Received  January 2017 Revised  July 2017 Published  June 2018

Fund Project: This work is supported in part by the National Natural Science Foundation of China (11501403; 11461023; 11701322; 11561072) and the Shanxi Province Science Foundation for Youths under grant 2013021001-3 and the Honghe University Doctoral Research Programs (XJ17B11, XJ17B12) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013).

In this paper, we study the following Kirchhoff-type equation with critical growth
 $-(a+b\int {_{\mathbb{R}^3}} |\nabla u|^2dx)\triangle u+V(x)u = λ f(x,u)+|u|^4u, \; x \; ∈\mathbb{R}^3,$
where a>0, b>0, λ>0 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as
 $b\searrow 0$
.
Citation: 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
##### References:
 [1] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $R^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. [2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2. [3] S. Bernstein, Sur une classe d'équations fonctionnells aux dérivées partielles, Bull. Acad. Sci. URSS, Sér. Math. [Izv. Akad. Nauk SSSR], 4 (1940), 17-26. [4] T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, In Stationary Partial Differential Equations. Handb. Differ. Equ., vol. Ⅱ, pp. 1–55. Elsevier/North-Holland, Amsterdam (2005). doi: 10.1016/S1874-5733(05)80009-9. [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. [6] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. [7] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7. [8] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. [9] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605. [10] G. M. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non-homogeneous operators, Indiana University Mathematics Journal, 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828. [11] Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902. doi: 10.1016/j.jde.2015.04.005. [12] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [13] X. He and W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500. doi: 10.1007/s10231-012-0286-6. [14] L. Jeanjean, On the existence of bounded Palais -Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [16] J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud., 30 (1978), 284–346. [17] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser. ), 46 (2003), 625–630 (in Chinese). [18] Q. Li and X. Wu, A new result on high energy solutions for Schrödinger-Kirchhoff type equations in $\mathbb{R}^N$, Appl. Math. Lett., 30 (2014), 24-27. doi: 10.1016/j.aml.2013.12.002. [19] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [20] S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.), 96 (1975), 152-168. [21] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. [22] A. Szulkin, T. Weth, The method of Nehari manifold, in D. Y. Gao, D. Motreanu (Eds), Handbook of Nonconvex Analysis and Applications, International Press, Boston, (2010), 597– 632. [23] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N$, Nonlinear Anal., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. [24] X. Wu, High energy solutions of systems of Kirchhoff-type equations in $\mathbb{R}^N$, J. Math. Phy., 53 (2012), 063508. doi: 10.1063/1.4729543. [25] X. Wu and K. Wu, Geometrically distinct solutions for quasilinear elliptic equations, Nonlinearity, 27 (2014), 987-1001. [26] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023.

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##### References:
 [1] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $R^N$, Nonlinear Anal., 75 (2012), 2750-2759. doi: 10.1016/j.na.2011.11.017. [2] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2. [3] S. Bernstein, Sur une classe d'équations fonctionnells aux dérivées partielles, Bull. Acad. Sci. URSS, Sér. Math. [Izv. Akad. Nauk SSSR], 4 (1940), 17-26. [4] T. Bartsch, Z. Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, In Stationary Partial Differential Equations. Handb. Differ. Equ., vol. Ⅱ, pp. 1–55. Elsevier/North-Holland, Amsterdam (2005). doi: 10.1016/S1874-5733(05)80009-9. [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and uniform decay for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701-730. [6] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. [7] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7. [8] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. [9] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605. [10] G. M. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non-homogeneous operators, Indiana University Mathematics Journal, 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828. [11] Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations, 259 (2015), 2884-2902. doi: 10.1016/j.jde.2015.04.005. [12] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [13] X. He and W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500. doi: 10.1007/s10231-012-0286-6. [14] L. Jeanjean, On the existence of bounded Palais -Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [16] J. L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1997, in: North-Holland Math. Stud., 30 (1978), 284–346. [17] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser. ), 46 (2003), 625–630 (in Chinese). [18] Q. Li and X. Wu, A new result on high energy solutions for Schrödinger-Kirchhoff type equations in $\mathbb{R}^N$, Appl. Math. Lett., 30 (2014), 24-27. doi: 10.1016/j.aml.2013.12.002. [19] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [20] S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.), 96 (1975), 152-168. [21] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040. [22] A. Szulkin, T. Weth, The method of Nehari manifold, in D. Y. Gao, D. Motreanu (Eds), Handbook of Nonconvex Analysis and Applications, International Press, Boston, (2010), 597– 632. [23] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N$, Nonlinear Anal., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023. [24] X. Wu, High energy solutions of systems of Kirchhoff-type equations in $\mathbb{R}^N$, J. Math. Phy., 53 (2012), 063508. doi: 10.1063/1.4729543. [25] X. Wu and K. Wu, Geometrically distinct solutions for quasilinear elliptic equations, Nonlinearity, 27 (2014), 987-1001. [26] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023.
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