# American Institute of Mathematical Sciences

May  2020, 19(5): 2819-2838. doi: 10.3934/cpaa.2020123

## Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term

 1 School of Mathematical Sciences, Nankai University, Tianjin, 300071, China 2 School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P. R. China

* Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: This work was supported by NNSF of China (11471170, 11790271), Innovation and development project of Guangzhou University, Nankai Zhide Foundation, Tianjin postgraduate researth and innovation project(2019YJSB041)

In this paper, we consider the following elliptic problem
 $-\texttt{div}(|\nabla u|^{N-2}\nabla u)+V(x)|u|^{N-2}u = \frac{f(x, u)}{|x|^{\eta}}\; \; \operatorname{in}\; \; \mathbb{R}^{N}$
and its perturbation problem, where
 $N\geq 2$
,
 $0<\eta , $ V(x) \geq V_{0 }> 0 $and $ f(x, t) $has a critical exponential growth behavior. By using the variational technique and the indirection method, the existence of a positive ground state solution is proved. For the perturbation problem, the existence of two distinct nontrivial weak solutions is proved. Citation: Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123 ##### References:  [1] A. Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in$\mathbb{R}^{N}$and its applications, Int. Math. Res. Notices, 13 (2010), 2394-2426. doi: 10.1093/imrn/rnp194. [2] C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in$\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1288-1311. doi: 10.1016/j.jde.2008.08.004. [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [4] D. M. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in$\mathbb{R}^{2}$, Commun. Partial Differ. Equ., 17 (1992), 407-435. doi: 10.1080/03605309208820848. [5] G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. [6] G. Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl., 124 (1980), 161-179. doi: 10.1007/BF01795391. [7] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in$\mathbb{R}^2$with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 3 (1995), 139-153. doi: 10.1007/BF01205003. [8] M. de Souza and J. M. do Ó, On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101. doi: 10.1007/s11118-012-9308-7. [9] J. M. do Ó, N-Laplacian equations in$\mathbb{R}^{N}$with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. [10] J. M. do Ó, E. de Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in$\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1363-1386. doi: 10.1016/j.jde.2008.11.020. [11] J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo, An improvement for the Trudinger-Moser inequality and applications, J. Differ. Equ., 256 (2014), 1317-1349. doi: 10.1016/j.jde.2013.10.016. [12] N. Lam and G. Lu, N-Laplacian equations in$\mathbb{R}^{N}$with subcritical and critical growth without the Ambrosetti-Rabinowitz condition, Adv. Nonlinear Stud., 13 (2013), 289-308. doi: 10.1515/ans-2013-0203. [13] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of$n$-Laplacian type with critical exponential growth in$\mathbb{R}^{N}$, J. Funct. Anal., 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012. [14] N. Lam and G. Lu, Elliptic equations and systems with subscritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143. doi: 10.1007/s12220-012-9330-4. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam, 1 (1985), 145-201. doi: 10.4171/RMI/6. [16] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in$\mathbb{R}^{N}$, Proc. Indian Acad. Sci. Math. Sci., 105 (1995), 425-444. doi: 10.1007/BF02836879. [17] Y. Yang, Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differ. Equ., 252 (2012), 2266-2295. doi: 10.1016/j.jde.2011.08.027. [18] Y. Yang, Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704. doi: 10.1016/j.jfa.2011.11.018. [19] J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo, Critical points for a functional involving critical growth of Trudinger-Moser type, Potential Anal., 42 (2015), 229-246. doi: 10.1007/s11118-014-9431-8. [20] C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in$\mathbb{R}^{N}$and$n$-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585. doi: 10.1515/ans-2017-6041. show all references ##### References:  [1] A. Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in$\mathbb{R}^{N}$and its applications, Int. Math. Res. Notices, 13 (2010), 2394-2426. doi: 10.1093/imrn/rnp194. [2] C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in$\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1288-1311. doi: 10.1016/j.jde.2008.08.004. [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [4] D. M. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in$\mathbb{R}^{2}$, Commun. Partial Differ. Equ., 17 (1992), 407-435. doi: 10.1080/03605309208820848. [5] G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. [6] G. Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl., 124 (1980), 161-179. doi: 10.1007/BF01795391. [7] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in$\mathbb{R}^2$with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 3 (1995), 139-153. doi: 10.1007/BF01205003. [8] M. de Souza and J. M. do Ó, On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents, Potential Anal., 38 (2013), 1091-1101. doi: 10.1007/s11118-012-9308-7. [9] J. M. do Ó, N-Laplacian equations in$\mathbb{R}^{N}$with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. [10] J. M. do Ó, E. de Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in$\mathbb{R}^{N}$, J. Differ. Equ., 246 (2009), 1363-1386. doi: 10.1016/j.jde.2008.11.020. [11] J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo, An improvement for the Trudinger-Moser inequality and applications, J. Differ. Equ., 256 (2014), 1317-1349. doi: 10.1016/j.jde.2013.10.016. [12] N. Lam and G. Lu, N-Laplacian equations in$\mathbb{R}^{N}$with subcritical and critical growth without the Ambrosetti-Rabinowitz condition, Adv. Nonlinear Stud., 13 (2013), 289-308. doi: 10.1515/ans-2013-0203. [13] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of$n$-Laplacian type with critical exponential growth in$\mathbb{R}^{N}$, J. Funct. Anal., 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012. [14] N. Lam and G. Lu, Elliptic equations and systems with subscritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143. doi: 10.1007/s12220-012-9330-4. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam, 1 (1985), 145-201. doi: 10.4171/RMI/6. [16] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in$\mathbb{R}^{N}$, Proc. Indian Acad. Sci. Math. Sci., 105 (1995), 425-444. doi: 10.1007/BF02836879. [17] Y. Yang, Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differ. Equ., 252 (2012), 2266-2295. doi: 10.1016/j.jde.2011.08.027. [18] Y. Yang, Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704. doi: 10.1016/j.jfa.2011.11.018. [19] J. M. do Ó, M. de Souza, E. de Medeiros and U. Severo, Critical points for a functional involving critical growth of Trudinger-Moser type, Potential Anal., 42 (2015), 229-246. doi: 10.1007/s11118-014-9431-8. [20] C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger-Moser inequalities in$\mathbb{R}^{N}$and$n$-Laplace equations, Adv. Nonlinear Stud., 18 (2018), 567-585. doi: 10.1515/ans-2017-6041.  [1] Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. 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