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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. Berestycki, I. 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 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., 4 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar [8] A. Dillon, P. 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 Ó, F. 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. Filippucci, P. 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. Zhang, S. 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

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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. Berestycki, I. 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 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., 4 (1995), 139-153.  doi: 10.1007/BF01205003.  Google Scholar [8] A. Dillon, P. 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 Ó, F. 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. Filippucci, P. 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. Zhang, S. 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
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