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November  2019, 18(6): 3217-3242. doi: 10.3934/cpaa.2019145

## The effect of nonlocal term on the superlinear elliptic equations in $\mathbb{R}^{N}$

 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China 2 School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China 3 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

* Corresponding author

Received  December 2018 Revised  February 2019 Published  May 2019

Fund Project: J. Sun was supported by the National Natural Science Foundation of China (Grant No. 11671236). T. F. Wu was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 106-2115-M-390-001-MY2) and the National Center for Theoretical Sciences, Taiwan.

We are concerned with a class of nonlocal elliptic equations as follows:
 \left\{ \begin{align} & -M\left( \int_{{{\mathbb{R}}^{N}}}{|}\nabla u{{|}^{2}}dx \right)\Delta u+\lambda V\left( x \right)u=f(x,u)\ \ \ \ \text{in }{{\mathbb{R}}^{N}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{N}}), \\ \end{align} \right.
where
 $N\geq 1,$
 $\lambda>0$
is a parameter,
 $M(t) = am(t)+b$
with
 $a, b>0$
and
 $m\in C(\mathbb{R}^{+}, \mathbb{R}^{+})$
,
 $V\in C(\mathbb{R}^{N}, \mathbb{R}^{+})$
and
 $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$
satisfying
 $\lim_{|u|\rightarrow \infty }f(x, u) /|u|^{k-1} = q(x)$
uniformly in
 $x\in \mathbb{R}^{N}$
for any
 $2 ( $ 2^{\ast} = \infty $for $ N = 1, 2 $and $ 2^{\ast} = 2N/(N-2) $for $ N\geq 3 $). Unlike most other papers on this problem, we are more interested in the effects of the functions $ m $and $ q $on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem. Citation: Juntao Sun, Tsung-fang Wu. The effect of nonlocal term on the superlinear elliptic equations in$ \mathbb{R}^{N} $. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3217-3242. doi: 10.3934/cpaa.2019145 ##### References:  [1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. [2] C. O. Alves and G. M. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26. doi: 10.1515/anona-2015-0101. [3] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on$\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [5] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/s000330050003. [6] A. Bensedki and M. Bouchekif, On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity, Math. Comp. Model., 49 (2009), 1089-1096. doi: 10.1016/j.mcm.2008.07.032. [7] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 8 (1983), 486-490. doi: 10.2307/2044999. [8] C. Chen, Y. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. [9] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in$\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. [10] Y. Deng and W. Shuai, Sign-changing multi-bump solutions for Kirchhoff type equations in$\mathbb{R}^{3}$., Discrete Continuous Dynam. Systems - A, 38 (2018), 3139-3168. doi: 10.3934/dcds.2018137. [11] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3. [12] G. M. Figueiredo, N. Ikoma and J. R. S. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [13] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in$\mathbb{R}^{3}$involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. [14] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in$\mathbb{R}^{3}$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [15] N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials, Discrete Continuous Dynam. Systems - A, 35 (2015), 943-966. doi: 10.3934/dcds.2015.35.943. [16] T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1. [17] A. Mao and H. Chang, Kirchhoff type problems in$\mathbb{R}^{N}$with radial potentials and locally Lipschitz functional, Appl. Math. Lett., 62 (2016), 49-54. doi: 10.1016/j.aml.2016.06.014. [18] A. Mao and S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021. [19] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193. doi: 10.1016/j.jde.2014.05.002. [20] 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. [21] J. Sun and T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792. doi: 10.1016/j.jde.2013.12.006. [22] J. Sun and T. F. Wu, On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669. doi: 10.1007/s00033-015-0494-1. [23] J. Sun and T. F. Wu, Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 435-448. doi: 10.1017/S0308210515000475. [24] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invarint 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. Alves, F. J. S. A. Corrêa 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] C. O. Alves and G. M. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26. doi: 10.1515/anona-2015-0101. [3] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. [4] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on$\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [5] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/s000330050003. [6] A. Bensedki and M. Bouchekif, On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity, Math. Comp. Model., 49 (2009), 1089-1096. doi: 10.1016/j.mcm.2008.07.032. [7] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 8 (1983), 486-490. doi: 10.2307/2044999. [8] C. Chen, Y. Kuo and T. F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017. [9] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in$\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527. doi: 10.1016/j.jfa.2015.09.012. [10] Y. Deng and W. Shuai, Sign-changing multi-bump solutions for Kirchhoff type equations in$\mathbb{R}^{3}$., Discrete Continuous Dynam. Systems - A, 38 (2018), 3139-3168. doi: 10.3934/dcds.2018137. [11] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3. [12] G. M. Figueiredo, N. Ikoma and J. R. S. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [13] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in$\mathbb{R}^{3}$involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. [14] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in$\mathbb{R}^{3}$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [15] N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials, Discrete Continuous Dynam. Systems - A, 35 (2015), 943-966. doi: 10.3934/dcds.2015.35.943. [16] T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1. [17] A. Mao and H. Chang, Kirchhoff type problems in$\mathbb{R}^{N}$with radial potentials and locally Lipschitz functional, Appl. Math. Lett., 62 (2016), 49-54. doi: 10.1016/j.aml.2016.06.014. [18] A. Mao and S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021. [19] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (2014), 1168-1193. doi: 10.1016/j.jde.2014.05.002. [20] 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. [21] J. Sun and T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792. doi: 10.1016/j.jde.2013.12.006. [22] J. Sun and T. F. Wu, On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669. doi: 10.1007/s00033-015-0494-1. [23] J. Sun and T. F. Wu, Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 435-448. doi: 10.1017/S0308210515000475. [24] Z. Zhang and K. 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