Berchio, Elvise: 
A Note on Some Nonlinear Fourth Order Differential Equations
 Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.2, p. 349-361,  (English)
pdf (744 Kb), djvu (132 Kb).  | MR 3112983  | Zbl 1291.34031 
Sunto
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).
Referenze Bibliografiche
[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. | 
fulltext (doi) | 
MR 2269942 | 
Zbl 1152.35360[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. | 
fulltext (doi) | 
MR 2139208 | 
Zbl 1162.35339[3] 
E. BERCHIO - 
D. CASSANI - 
F. GAZZOLA, 
Hardy-Rellich inequalities with boundary remainder terms and applications, 
Manuscripta Math. 131 (
2010), 427-458. | 
fulltext (doi) | 
MR 2592089 | 
Zbl 1187.35045[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. | 
fulltext (doi) | 
MR 2860632 | 
Zbl 1235.35126[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. | 
fulltext (doi) | 
MR 2831710 | 
Zbl 1236.34042[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 2135245 | 
Zbl 1129.35349[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. | 
fulltext (doi) | 
MR 2457633[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. | 
fulltext (doi) | 
MR 2220064 | 
Zbl 1104.34034[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. | 
fulltext (doi) | 
MR 1448828 | 
Zbl 0879.35113[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. | 
fulltext (doi) | 
MR 2338421 | 
Zbl 1138.35022[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. | 
fulltext (doi) | 
MR 2569861 | 
Zbl 1190.34017[13] 
F. GAZZOLA - 
R. PAVANI, 
Blow up oscillating solutions to some nonlinear fourth order differential equations, 
Nonlin. Anal. TMA 74 (
2011), 6696-6711. | 
fulltext (doi) | 
MR 2834070 | 
Zbl 1237.34043[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 110868[18] 
P. KARAGEORGIS - 
J. G. STALKER, 
A lower bound for the amplitude of traveling waves of suspension bridges, 
Nonlinear Analysis 75 (
2012), 5212-5214. | 
fulltext (doi) | 
MR 2927583 | 
Zbl 1266.34078[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. | 
fulltext (doi) | 
MR 1611691 | 
Zbl 0933.35057[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. | 
fulltext (doi) | 
MR 2278730 | 
Zbl 1117.35025[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). | 
fulltext (doi) | 
MR 1839555 | 
Zbl 1076.34515[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. | 
fulltext (doi) | 
MR 1929147 | 
Zbl 1029.34036