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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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