A Note on Some Nonlinear Fourth Order Differential Equations
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 349-361.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

For a family of fourth order semilinear ordinary differential equations we discuss some fundamental issues, such as global continuation of solutions and their qualitative behavior. The note is the summary of a communication given at the XIX Congress of U.M.I. (Bologna - September 12-17, 2011).
@article{BUMI_2013_9_6_2_a4,
     author = {Berchio, Elvise},
     title = {A {Note} on {Some} {Nonlinear} {Fourth} {Order} {Differential} {Equations}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {349--361},
     publisher = {mathdoc},
     volume = {Ser. 9, 6},
     number = {2},
     year = {2013},
     zbl = {1291.34031},
     mrnumber = {3112983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a4/}
}
TY  - JOUR
AU  - Berchio, Elvise
TI  - A Note on Some Nonlinear Fourth Order Differential Equations
JO  - Bollettino della Unione matematica italiana
PY  - 2013
SP  - 349
EP  - 361
VL  - 6
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a4/
LA  - en
ID  - BUMI_2013_9_6_2_a4
ER  - 
%0 Journal Article
%A Berchio, Elvise
%T A Note on Some Nonlinear Fourth Order Differential Equations
%J Bollettino della Unione matematica italiana
%D 2013
%P 349-361
%V 6
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a4/
%G en
%F BUMI_2013_9_6_2_a4
Berchio, Elvise. A Note on Some Nonlinear Fourth Order Differential Equations. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 349-361. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a4/

[1] G. Arioli - F. Gazzola - H.-Ch. Grunau, Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity, J. Diff. Eq. 230 (2006), 743-770. | DOI | MR | Zbl

[2] G. Arioli - F. Gazzola - H.-Ch. Grunau - E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36, (2005), 1226-1258. | DOI | MR | Zbl

[3] E. Berchio - D. Cassani - F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications, Manuscripta Math. 131 (2010), 427-458. | DOI | MR | Zbl

[4] E. Berchio - A. Farina - A. Ferrero - F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Diff. Eq. 252 (2012), 2596-2616. | DOI | MR | Zbl

[5] E. Berchio - A. Ferrero - F. Gazzola - P. Karageorgis, Qualitative behavior of global solutions to some nonlinear fourth order differential equations, J. Diff. Eq. 251 (2011), 2696-2727. | DOI | MR | Zbl

[6] E. Berchio - F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. Diff. Eq. 34 (2005), 1-20. | fulltext EuDML | MR | Zbl

[7] D. Bonheure - L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Diff. Eq. Vol. III, Elsevier Science (2006), 103-202. | DOI | MR

[8] B. Breuer - J. Hórak - P. J. Mckenna - M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq. 224 (2006), 60-97. | DOI | MR | Zbl

[9] S. Y. A. Chang - W. Chen, A note on a class of higher order conformally covariant equations, Discrete Cont. Dyn. Syst. 7 (2001), 275-281. | DOI | MR | Zbl

[10] Y. Chen - P. J. Mckenna, Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations, J. Diff. Eq. 136 (1997), 325-355. | DOI | MR | Zbl

[11] J. Dávila - L. Dupaigne - I. Guerra - M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal. 39 (2007), 565-592. | DOI | MR | Zbl

[12] J. Dávila - I. Flores - I. Guerra, Multiplicity of solutions for a fourth order problem with exponential nonlinearity, J. Diff. Eq. 247 (2009), 3136-3162. | DOI | MR | Zbl

[13] F. Gazzola - R. Pavani, Blow up oscillating solutions to some nonlinear fourth order differential equations, Nonlin. Anal. TMA 74 (2011), 6696-6711. | DOI | MR | Zbl

[14] F. Gazzola - R. Pavani, Blow-up oscillating solutions to some nonlinear fourth order differential equations describing oscillations of suspension bridges, IAB- MAS12, 6th International Conference on Bridge Maintenance, Safety, Management, Resilience and Sustainability, Stresa 2012, Biondini-Frangopol (Editors), Taylor-Francis Group (London, 2001), 3089-3093.

[15] I. M. Gel'Fand, Some problems in the theory of quasilinear equations, Section 15, due to G. I. Barenblatt, Amer. Math. Soc. Transl. II. Ser. 29 (1963), 295-381. Russian original: Uspekhi Mat. Nauk. 14 (1959), 87-158. | MR

[16] D. Joseph - T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal. 49 (1973), 241-269. | DOI | MR | Zbl

[17] P. Karageorgis - P. J. Mckenna, The existence of ground states for fourth-order wave equations, Nonlinear Analysis 73 (2010), 367-373. | DOI | MR | Zbl

[18] P. Karageorgis - J. G. Stalker, A lower bound for the amplitude of traveling waves of suspension bridges, Nonlinear Analysis 75 (2012), 5212-5214. | DOI | MR | Zbl

[19] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^n$, Comment. Math. Helv. 73 (1998), 206-231. | DOI | MR | Zbl

[20] P. J. Mckenna - W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990), 703-715. | DOI | MR | Zbl

[21] P. J. Mckenna, Large-amplitude periodic oscillations in simple and complex mechanical systems: outgrowths from nonlinear analysis, Milan J. Math. 74 (2006), 79-115. | DOI | MR | Zbl

[22] L. A. Peletier - W. C. Troy, Spatial patterns. Higher order models in physics and mechanics. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston Inc., Boston, MA 45 (2001). | DOI | MR | Zbl

[23] M. A. Peletier, Sequential buckling: a variational analysis, SIAM J. Math. Anal. 32 (2001), 1142-1168. | DOI | MR | Zbl

[24] D. Smets - J. B. Van Den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Diff. Eq. 184 (2002), 78-96. | DOI | MR | Zbl