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November  2015, 14(6): 2185-2201. doi: 10.3934/cpaa.2015.14.2185

On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth

1. 

Institut Supérieur des Mathématiques Appliquées et de l'Informatique de Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan , Tunisia

Received  October 2014 Revised  August 2015 Published  September 2015

In this paper, we prove a multiplicity result for some semilinear elliptic equation of biharmoninc type in $R^4$ containing a Laplacian term. The nonlinear term exhibits an exponential growth.
Citation: Sami Aouaoui. On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2185-2201. doi: 10.3934/cpaa.2015.14.2185
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, {\em J. Funct. Anal.}, 14 (1973), 349. Google Scholar

[2]

J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $ \mathbbR^N, $, {\em Nonlinear Anal.}, 49 (2002), 861. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar

[3]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear ellipitc PDE on $R^n$,, {\em }, 45 (1992), 1217. doi: 10.1002/cpa.3160451002. Google Scholar

[4]

Y. Deng and Y. Li, Regularity of the solutions for nonlinear biharmonic equations in $ \mathbbR^N, $, {\em Acta. Math. Sci.}, 29 (2009), 1469. doi: 10.1016/S0252-9602(09)60119-3. Google Scholar

[5]

I. Ekeland, On the variational principle,, {\em J. Math. Anal. App.}, 47 (1974), 324. Google Scholar

[6]

D. G. de Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques,, {\em Differ. Integral Equ.}, 17 (2004), 119. Google Scholar

[7]

L. R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth,, {\em Nonlinear Anal.}, 95 (2014), 607. doi: 10.1016/j.na.2013.10.010. Google Scholar

[8]

O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer-verlag, (1993). Google Scholar

[9]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth,, {\em Discrete Contin. Dyn. Syst.}, 32 (2012), 2187. doi: 10.3934/dcds.2012.32.2187. Google Scholar

[10]

A. C. Lazer and P. J. Mckenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, {\em SIAM Rev.}, 32 (1990), 537. doi: 10.1137/1032120. Google Scholar

[11]

CH. Li and C-L. Tang, Three solutions for a Navier boundary value problem involving the $p$-biharmonic,, {\em Nonlinear Anal.}, 72 (2010), 1339. doi: 10.1016/j.na.2009.08.011. Google Scholar

[12]

M. T. Pimenta and S. H. Soares, Existence and concentration of solutions for a class of biharmonic equations,, {\em J. Math. Anal. Appl.}, 390 (2012), 274. doi: 10.1016/j.jmaa.2012.01.039. Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, {\em Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[14]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR^N, $, {\em Trans. Amer. Math. Soc.}, 365 (2013), 645. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[15]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with critical exponential growth,, {\em Commun. Pure Appl. Anal.}, 12 (2013), 405. doi: 10.3934/cpaa.2013.12.405. Google Scholar

[16]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with subcritical exponential growth,, {\em Adv. Nonlinear Stud.}, 11 (2011), 889. Google Scholar

[17]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation,, {\em J. Differential Equations}, 246 (2009), 3109. doi: 10.1016/j.jde.2009.02.016. Google Scholar

[18]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type,, {\em J. Math. Anal. Appl.}, 348 (2008), 730. doi: 10.1016/j.jmaa.2008.07.068. Google Scholar

[19]

M. Willem, Minimax Theorem,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[20]

F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation,, {\em Nonlinear Anal.}, 70 (2009), 1365. doi: 10.1016/j.na.2008.02.016. Google Scholar

[21]

Y. Yang and J. Zhang, Existence of solutions for some fourth-order nonlinear elliptic problems,, {\em J. Math. Anal. Appl.}, 351 (2009), 128. doi: 10.1016/j.jmaa.2008.08.023. Google Scholar

[22]

Y. Yang, Adams type inequalities and related ellipitc partial differential equations in dimension four,, {\em J. Differential Equations}, 252 (2012), 2266. doi: 10.1016/j.jde.2011.08.027. Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, {\em J. Funct. Anal.}, 14 (1973), 349. Google Scholar

[2]

J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $ \mathbbR^N, $, {\em Nonlinear Anal.}, 49 (2002), 861. doi: 10.1016/S0362-546X(01)00144-4. Google Scholar

[3]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear ellipitc PDE on $R^n$,, {\em }, 45 (1992), 1217. doi: 10.1002/cpa.3160451002. Google Scholar

[4]

Y. Deng and Y. Li, Regularity of the solutions for nonlinear biharmonic equations in $ \mathbbR^N, $, {\em Acta. Math. Sci.}, 29 (2009), 1469. doi: 10.1016/S0252-9602(09)60119-3. Google Scholar

[5]

I. Ekeland, On the variational principle,, {\em J. Math. Anal. App.}, 47 (1974), 324. Google Scholar

[6]

D. G. de Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques,, {\em Differ. Integral Equ.}, 17 (2004), 119. Google Scholar

[7]

L. R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth,, {\em Nonlinear Anal.}, 95 (2014), 607. doi: 10.1016/j.na.2013.10.010. Google Scholar

[8]

O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer-verlag, (1993). Google Scholar

[9]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth,, {\em Discrete Contin. Dyn. Syst.}, 32 (2012), 2187. doi: 10.3934/dcds.2012.32.2187. Google Scholar

[10]

A. C. Lazer and P. J. Mckenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, {\em SIAM Rev.}, 32 (1990), 537. doi: 10.1137/1032120. Google Scholar

[11]

CH. Li and C-L. Tang, Three solutions for a Navier boundary value problem involving the $p$-biharmonic,, {\em Nonlinear Anal.}, 72 (2010), 1339. doi: 10.1016/j.na.2009.08.011. Google Scholar

[12]

M. T. Pimenta and S. H. Soares, Existence and concentration of solutions for a class of biharmonic equations,, {\em J. Math. Anal. Appl.}, 390 (2012), 274. doi: 10.1016/j.jmaa.2012.01.039. Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, {\em Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[14]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR^N, $, {\em Trans. Amer. Math. Soc.}, 365 (2013), 645. doi: 10.1090/S0002-9947-2012-05561-9. Google Scholar

[15]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with critical exponential growth,, {\em Commun. Pure Appl. Anal.}, 12 (2013), 405. doi: 10.3934/cpaa.2013.12.405. Google Scholar

[16]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with subcritical exponential growth,, {\em Adv. Nonlinear Stud.}, 11 (2011), 889. Google Scholar

[17]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation,, {\em J. Differential Equations}, 246 (2009), 3109. doi: 10.1016/j.jde.2009.02.016. Google Scholar

[18]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type,, {\em J. Math. Anal. Appl.}, 348 (2008), 730. doi: 10.1016/j.jmaa.2008.07.068. Google Scholar

[19]

M. Willem, Minimax Theorem,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[20]

F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation,, {\em Nonlinear Anal.}, 70 (2009), 1365. doi: 10.1016/j.na.2008.02.016. Google Scholar

[21]

Y. Yang and J. Zhang, Existence of solutions for some fourth-order nonlinear elliptic problems,, {\em J. Math. Anal. Appl.}, 351 (2009), 128. doi: 10.1016/j.jmaa.2008.08.023. Google Scholar

[22]

Y. Yang, Adams type inequalities and related ellipitc partial differential equations in dimension four,, {\em J. Differential Equations}, 252 (2012), 2266. doi: 10.1016/j.jde.2011.08.027. Google Scholar

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