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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 and 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. [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. [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. [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. [5] I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353. [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. [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. [8] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques Springer-verlag, Heidelberg, 1993. [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. [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. [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. [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. [13] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [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. [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. [16] F. Sani, A biharmonic equation in $\mathbbR^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904. [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. [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. [19] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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.

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. [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. [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. [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. [5] I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353. [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. [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. [8] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques Springer-verlag, Heidelberg, 1993. [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. [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. [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. [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. [13] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [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. [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. [16] F. Sani, A biharmonic equation in $\mathbbR^4$ involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud., 11 (2011), 889-904. [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. [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. [19] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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.
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