Fourth-order elliptic problems involving concave-superlinear nonlinearities. (English) Zbl 1512.35225

Summary: The existence of solutions for a huge class of superlinear elliptic problems involving fourth-order elliptic problems defined on bounded domains under Navier boundary conditions is established. To this end we do not apply the well-known Ambrosetti-Rabinowitz condition. Instead, we assume that the nonlinear term is nonquadratic at infinity. Furthermore, the nonlinear term is a concave-superlinear function which can be indefinite in sign. In order to apply variational methods we employ some delicate arguments recovering some kind of compactness.


35J40 Boundary value problems for higher-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
Full Text: DOI Link


[1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[2] S. Agmon, A. Douglis and L. Niremberg, Estimates near the boundary for elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623-727. · Zbl 0093.10401
[3] C.O. Alves and J.M. do O, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud. 2 (2002), 437-458. · Zbl 1290.35067
[4] E. Berchio, A. Ferrero, F. Gazzola and P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations, J. Differential Equations 251 (2011), 2696-2727. · Zbl 1236.34042
[5] E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations 229 (2006), 1-23. · Zbl 1142.35016
[6] E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary condition, Adv. Differential Equations 12 (2007), no. 4, 381-406. · Zbl 1155.35018
[7] F. Bernis, J. Garcia Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth-order, Adv. Differential Equations 1 (1996), 219-240. · Zbl 0841.35036
[8] K.J. Brown and T.F. Wu, A Fibering map approach to a semilinear elliptic boundary value problem, Electronic J. Differential Equations 69 (2007), 1-9. · Zbl 1133.35337
[9] J. Chabrowski and J.M. do O, On some fourth-order semilinear elliptic problems in \(\mathbb R^N\), Nonlinear Anal. 49 (2002), 861-884. · Zbl 1011.35045
[10] Y. Chen and P.J. McKenna, Traveling waves in a nonlinear suspension beam: theoretical results and numerical observations, J. Differential Equations 135 (1997), 325-355. · Zbl 0879.35113
[11] F.J.S.A. Correa, J.V. Goncalves and A. Roncalli, On a class of fourth order nonlinear elliptic equations under Navier boundary conditions, Anal. Appl. (Singap.) 8 (2010), 185-197. · Zbl 1194.35163
[12] D.G. Costa, C.A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal. 23 (1994), 1401-1412. · Zbl 0820.35059
[13] E.D. da Silva and T.R. Cavalcante, Multiplicity of solutions to fourth-order superlinear elliptic problems under Navier conditions, Electron. J. Difer. Equ. 167 (2017), 1-16. · Zbl 1370.35096
[14] G. Figueiredo, M.F. Furtado and J.P. Silva, Existence and multiplicity of positive solutions for a fourth-order elliptic equation, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 2, 1053-1069. · Zbl 1437.35387
[15] M. Furtado and E. da Silva, Superlinear elliptic problems under the nonquadriticty condition at infinity, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 4, 779-790. · Zbl 1331.35113
[16] F. Gazzola, H.C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. · Zbl 1239.35002
[17] F. Gazzola and R. Pavani, Wide oscillation finite time blow up for solutions to nonlinear fourth-order differential equations, Arch. Rational Mech. Anal. 207 (2013), 717-752. · Zbl 1278.34036
[18] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2001. · Zbl 1042.35002
[19] J.V.A. Goncalves, E.D. Silva and M.L. Silva, On positive solutions for a fourth-order asymptotically linear elliptic equation under Navier boundary conditions, J. Math. Anal. Appl. 384 (2011), 387-399. · Zbl 1228.35103
[20] C.P. Gupta and Y.C. Kwong, Biharmonic eigenvalue problems and \(L_p\) estimates, Internat. J. Math. Math. Sci. 13 (1990), no. 3, 469-480. · Zbl 0709.35035
[21] L. Iturriaga, S. Lorca and P. Ubilla, A quasilinear problem without the Ambrosetti-Rabinowitz-type condition, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 391-398. · Zbl 1194.35183
[22] L. Jeanjean, On the existence of bounded Palais-Smale sequences and an application to Landemann-Lazer type problem set \(\mathbb R^N\), Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 797-809. · Zbl 0935.35044
[23] 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. · Zbl 0725.73057
[24] C. Li, R.P. Agarwal and Z.-Q. Ou, Existence of three nontrivial solutions for a class of fourth-order elliptic equations, Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 331-344. · Zbl 1400.35138
[25] Z. Liu and Z.Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004), 653-574. · Zbl 1113.35048
[26] P.J. Mckenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Ration. Mech. Anal. 98 (1987), no. 2, 167-177. · Zbl 0676.35003
[27] P.J. Mckenna and W. Walter, Travelling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990), 703-715. · Zbl 0699.73038
[28] A.M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal. 34 (1998), 509-523. · Zbl 0929.35053
[29] O.H. Miyagaki and M.A.S. Souto, Supelinear problems without Ambrosetti-Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628-3638. · Zbl 1158.35400
[30] Y. Pu, X.P. Wu and C.L. Tang, Fourth-order Navier boundary value problem with combined nonlinearities, J. Math. Anal. Appl. 398 (2013), 798-813. · Zbl 1253.35125
[31] T. Riedel and P.K. Sahoo, Mean Value Theorems and Functional Equations, World Scientific Publishing Company, 1998. · Zbl 0980.39015
[32] G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations 5 (1992), no. 3, 561-565. · Zbl 0786.35060
[33] W. Wang, A. Zang and P. Zhao, Multiplicity of solutions for a class of fourth elliptic equations, Nonlinear Anal. 70 (2009), 4377-4385. · Zbl 1162.35355
[34] Z.Q. Wang, On a supelinear ellitic equation , Anal. Inst. H. Poincare Anal. Nonlineare 8 (1991), 43-57. · Zbl 0733.35043
[35] W. Zhang, B. Cheng, X. Tang and J. Zhang, Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type, Commun. Pure Appl. Anal. 15 (2016), no. 6, 2161-2177. · Zbl 1353.35150
[36] J.W. Zhou and X. Wu, Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl. 342 (2008), 542-558. · Zbl 1138.35335
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.