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