• Previous Article
    Periodic solutions of an age-structured epidemic model with periodic infection rate
  • CPAA Home
  • This Issue
  • Next Article
    Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation
October  2020, 19(10): 4937-4953. doi: 10.3934/cpaa.2020219

Semilinear elliptic problems involving exponential critical growth in the half-space

1. 

Universidade Federal da Paraíba, Departamento de Matemática, 58051-900, João Pessoa-PB, Brazil

2. 

Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília-DF, Brazil

* Corresponding author

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: All the authors were supported by CNPq/Brazil. The second author was also supported by FAP-DF/Brazil. The third author was also supported by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ)

We perform an weighted Sobolev space approach to prove a Trudinger-Moser type inequality in the upper half-space. As applications, we derive some existence and multiplicity results for the problem
$ \begin{cases} -\Delta u+h(x)|u|^{q-2}u = a(x) f(u), &\mbox{in } \mathbb{R}^2_+,\\ \dfrac{\partial u}{\partial \nu}+u = 0, &\mbox{on } \partial\mathbb{R}^2_+, \end{cases} $
under some technical condition on
$ a $
,
$ b $
and the the exponential nonlinearity
$ f $
. The ideas can also be used to deal with Neumann boundary conditions.
Citation: Diego D. Felix, Marcelo F. Furtado, Everaldo S. Medeiros. Semilinear elliptic problems involving exponential critical growth in the half-space. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4937-4953. doi: 10.3934/cpaa.2020219
References:
[1]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Moser-Trudinger in $R^n$ and its applications, Int. Math. Res. Not., 13 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[2]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[3]

H. BerestyckiI. Capuzzo Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Meth. Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.  Google Scholar

[4]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Commun. Partial Differ. Equ., 17 (1992), 407-435.  doi: 10.1080/03605309208820848.  Google Scholar

[5]

J. Chabrowski, Elliptic variational problems with indefinite nonlinearities, Topol. Meth. Nonlinear Anal., 9 (1997), 221-231.  doi: 10.12775/TMNA.1997.010.  Google Scholar

[6]

F. Cîrstea and V. Rǎdulescu, Existence and non-existence results for a quasilinear problem with nonlinear boundary condition, J. Math. Anal. Appl., 244 (2000), 169-183.  doi: 10.1006/jmaa.1999.6699.  Google Scholar

[7]

D. G. de FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 4 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[8]

A. DillonP. K. Maini and H. G. Othmer, Pattern formation in generalized Turing systems, I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393.  doi: 10.1007/BF00160165.  Google Scholar

[9]

J. M. B. do ÓF. Sani and J. Zhang, Stationary nonlinear Schrödinger equations in $\mathbb{R}^2$ with potentials vanishing at infinity, Ann. Mat. Pura Appl., 196 (2017), 363-393.  doi: 10.1007/s10231-016-0576-5.  Google Scholar

[10]

R. FilippucciP. Pucci and V. Rǎdulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Commun. Partial Differ. Equ., 33 (2008), 706-717.  doi: 10.1080/03605300701518208.  Google Scholar

[11]

M. Guzmán, Differentiation of Integrals in $\mathbb{R}^2$, in Lecture Notes in Mathematics, Vol. 481, Springer, Berlin, 1975.  Google Scholar

[12]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[13]

B. Opic and A. Kufner, Hardy-type inequalities, in Pitman Research Notes in Mathematics Series, Vol. 219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[14]

K. Pflüger, Compact traces in weighted Sobolev spaces, Analysis, 18 (1998), 65-83.  doi: 10.1524/anly.1998.18.1.65.  Google Scholar

[15]

K. Pflüger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electron. J. Differ. Equ., 10 (1998), 1-13.   Google Scholar

[16]

V. Rǎdulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.  Google Scholar

[17]

J. Simon, Regularité de la solution d'une equation non lineaire dans $\mathbb{R}^2$, in Lecture Notes in Mathematics, Vol. 665, Springer, Heidelberg, 1978.  Google Scholar

[18]

Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal., 263 (2012), 1894-1938.  doi: 10.1016/j.jfa.2012.06.019.  Google Scholar

[19]

Y. Yang and X. Zhu, A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space, J. Partial Differ. Equ., 26 (2013), 300-304.   Google Scholar

[20]

J. ZhangS. Li and X. Xue, Multiple solutions for a class of semilinear elliptic problems with Robin boundary condition, J. Math. Anal. Appl., 388 (2012), 435-442.  doi: 10.1016/j.jmaa.2011.09.066.  Google Scholar

show all references

References:
[1]

Adimurthi and Y. Yang, An interpolation of Hardy inequality and Moser-Trudinger in $R^n$ and its applications, Int. Math. Res. Not., 13 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194.  Google Scholar

[2]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[3]

H. BerestyckiI. Capuzzo Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Meth. Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.  Google Scholar

[4]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Commun. Partial Differ. Equ., 17 (1992), 407-435.  doi: 10.1080/03605309208820848.  Google Scholar

[5]

J. Chabrowski, Elliptic variational problems with indefinite nonlinearities, Topol. Meth. Nonlinear Anal., 9 (1997), 221-231.  doi: 10.12775/TMNA.1997.010.  Google Scholar

[6]

F. Cîrstea and V. Rǎdulescu, Existence and non-existence results for a quasilinear problem with nonlinear boundary condition, J. Math. Anal. Appl., 244 (2000), 169-183.  doi: 10.1006/jmaa.1999.6699.  Google Scholar

[7]

D. G. de FigueiredoO. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 4 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar

[8]

A. DillonP. K. Maini and H. G. Othmer, Pattern formation in generalized Turing systems, I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393.  doi: 10.1007/BF00160165.  Google Scholar

[9]

J. M. B. do ÓF. Sani and J. Zhang, Stationary nonlinear Schrödinger equations in $\mathbb{R}^2$ with potentials vanishing at infinity, Ann. Mat. Pura Appl., 196 (2017), 363-393.  doi: 10.1007/s10231-016-0576-5.  Google Scholar

[10]

R. FilippucciP. Pucci and V. Rǎdulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Commun. Partial Differ. Equ., 33 (2008), 706-717.  doi: 10.1080/03605300701518208.  Google Scholar

[11]

M. Guzmán, Differentiation of Integrals in $\mathbb{R}^2$, in Lecture Notes in Mathematics, Vol. 481, Springer, Berlin, 1975.  Google Scholar

[12]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[13]

B. Opic and A. Kufner, Hardy-type inequalities, in Pitman Research Notes in Mathematics Series, Vol. 219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[14]

K. Pflüger, Compact traces in weighted Sobolev spaces, Analysis, 18 (1998), 65-83.  doi: 10.1524/anly.1998.18.1.65.  Google Scholar

[15]

K. Pflüger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electron. J. Differ. Equ., 10 (1998), 1-13.   Google Scholar

[16]

V. Rǎdulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530.  doi: 10.1016/j.na.2011.01.037.  Google Scholar

[17]

J. Simon, Regularité de la solution d'une equation non lineaire dans $\mathbb{R}^2$, in Lecture Notes in Mathematics, Vol. 665, Springer, Heidelberg, 1978.  Google Scholar

[18]

Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal., 263 (2012), 1894-1938.  doi: 10.1016/j.jfa.2012.06.019.  Google Scholar

[19]

Y. Yang and X. Zhu, A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space, J. Partial Differ. Equ., 26 (2013), 300-304.   Google Scholar

[20]

J. ZhangS. Li and X. Xue, Multiple solutions for a class of semilinear elliptic problems with Robin boundary condition, J. Math. Anal. Appl., 388 (2012), 435-442.  doi: 10.1016/j.jmaa.2011.09.066.  Google Scholar

[1]

Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221

[2]

Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic & Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001

[3]

Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048

[4]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609

[5]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[6]

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

[7]

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

[8]

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

[9]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

[10]

Kei Fong Lam, Hao Wu. Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1847-1878. doi: 10.3934/dcds.2020096

[11]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393

[12]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[13]

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

[14]

Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

[15]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[16]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[17]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935

[18]

Sang-Gyun Youn. On the Sobolev embedding properties for compact matrix quantum groups of Kac type. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3341-3366. doi: 10.3934/cpaa.2020148

[19]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117

[20]

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

2019 Impact Factor: 1.105

Article outline

[Back to Top]