American Institute of Mathematical Sciences

Advanced
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

Yanjun Liu 1, and  Chungen Liu 2,,

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<N $
,
$ 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.
Keywords: The ground state solution, singular elliptic equations, critical exponential growth, singular Trudinger–Moser inequality.
Mathematics Subject Classification: Primary: 35J60, 35B33; Secondary: 46E30.
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

[1]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
[2]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[3]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3331-3355. doi: 10.3934/cpaa.2021108
[4]

Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021137
[5]

Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707
[6]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
[7]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[8]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452
[9]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[10]

Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859
[11]

Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201
[12]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378
[13]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054
[14]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[15]

Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417
[16]

Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155
[17]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[18]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
[19]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[20]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (262)
  • HTML views (96)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences