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