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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

Sami Aouaoui 1,

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.
Keywords: Biharmoninc operator, Adams-type inequality, Palais-Smale condition, critical exponential growth, Laplacian term, iterative scheme..
Mathematics Subject Classification: Primary: 35J30, 35J35, 35J61; Secondary: 35J91, 35D3.
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, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[2]

J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $ \mathbbR^N, $ Nonlinear Anal., 49 (2002), 861-884. 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 $R^n$, 29 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar

[4]

Y. Deng and Y. Li, Regularity of the solutions for nonlinear biharmonic equations in $ \mathbbR^N, $ Acta. Math. Sci., 29 (2009), 1469-1480. doi: 10.1016/S0252-9602(09)60119-3.  Google Scholar

[5]

I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353.  Google Scholar

[6]

D. G. de Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integral Equ., 17 (2004), 119-126.  Google Scholar

[7]

L. R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth, Nonlinear Anal., 95 (2014), 607-624. 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, Heidelberg, 1993.  Google Scholar

[9]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205. 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, SIAM Rev., 32 (1990), 537-578. 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, Nonlinear Anal., 72 (2010), 1339-1347. 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, J. Math. Anal. Appl., 390 (2012), 274-289. doi: 10.1016/j.jmaa.2012.01.039.  Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[14]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR^N, $ Trans. Amer. Math. Soc., 365 (2013), 645-670. 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, Commun. Pure Appl. Anal., 12 (2013), 405-428. doi: 10.3934/cpaa.2013.12.405.  Google Scholar

[16]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.  Google Scholar

[17]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125. doi: 10.1016/j.jde.2009.02.016.  Google Scholar

[18]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738. doi: 10.1016/j.jmaa.2008.07.068.  Google Scholar

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[20]

F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation, Nonlinear Anal., 70 (2009), 1365-1376. 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, J. Math. Anal. Appl., 351 (2009), 128-137. 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, J. Differential Equations, 252 (2012), 2266-2295. 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, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[2]

J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $ \mathbbR^N, $ Nonlinear Anal., 49 (2002), 861-884. 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 $R^n$, 29 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar

[4]

Y. Deng and Y. Li, Regularity of the solutions for nonlinear biharmonic equations in $ \mathbbR^N, $ Acta. Math. Sci., 29 (2009), 1469-1480. doi: 10.1016/S0252-9602(09)60119-3.  Google Scholar

[5]

I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353.  Google Scholar

[6]

D. G. de Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integral Equ., 17 (2004), 119-126.  Google Scholar

[7]

L. R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth, Nonlinear Anal., 95 (2014), 607-624. 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, Heidelberg, 1993.  Google Scholar

[9]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205. 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, SIAM Rev., 32 (1990), 537-578. 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, Nonlinear Anal., 72 (2010), 1339-1347. 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, J. Math. Anal. Appl., 390 (2012), 274-289. doi: 10.1016/j.jmaa.2012.01.039.  Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[14]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR^N, $ Trans. Amer. Math. Soc., 365 (2013), 645-670. 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, Commun. Pure Appl. Anal., 12 (2013), 405-428. doi: 10.3934/cpaa.2013.12.405.  Google Scholar

[16]

F. Sani, A biharmonic equation in $ \mathbbR^4 $ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904.  Google Scholar

[17]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125. doi: 10.1016/j.jde.2009.02.016.  Google Scholar

[18]

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of $p$-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738. doi: 10.1016/j.jmaa.2008.07.068.  Google Scholar

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[20]

F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation, Nonlinear Anal., 70 (2009), 1365-1376. 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, J. Math. Anal. Appl., 351 (2009), 128-137. 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, J. Differential Equations, 252 (2012), 2266-2295. doi: 10.1016/j.jde.2011.08.027.  Google Scholar

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