Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Mathematische Annalen

Volumn 348, Issue 1, 2010, Pages 143-193

  Dávila, Juan ^a   Flores, Isabel ^b   Guerra, Ignacio ^c  

^a UNIVERSITY OF CHILE   (Chile)

^b UNIVERSITY OF CONCEPCIÓN   (Chile)

^c UNIVERSIDAD DE SANTIAGO DE CHILE   (Chile)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77953913768     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00208-009-0476-8     Document Type: Article

References (39)

1. Arioli G., Gazzola F., Grunau H.-C.: Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity. J. Differ. Equ. 230(2), 743-770 (2006).
2. Arioli G., Gazzola F., Grunau H.-C., Mitidieri E.: A semilinear fourth order elliptic problem with exponential nonlinearity. SIAM J. Math. Anal. 36(4), 1226-1258 (2005).
3. Bamón R., Flores I., del Pino M.: Ground states of semilinear elliptic equations: a geometric approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(5), 551-581 (2000).
4. Belickiǐ G. R.: Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class. Funct. Anal. Appl. 7, 268-277 (1973).
5. Berchio, E., Gazzola, F.: Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities. Electron. J. Differ. Equ. vol. 34, p. 20 (2005).
6. Berchio E., Gazzola F., Weth T.: Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems. J. Reine Angew. Math. 620, 165-183 (2008).
7. Cassani D., do Ó J. M., Ghoussoub N.: On a fourth order elliptic problem with a singular nonlinearity. Adv. Nonlinear Stud. 9(1), 177-197 (2009).
8. Chicone, C.: Ordinary differential equations with applications, 2nd edn. Texts in Applied Mathematics, vol. 34. Springer, New York (2006).
9. Choi Y. S., Xu X.: Nonlinear biharmonic equations with negative exponents. J. Differ. Equ. 246(1), 216-234 (2009).
10. Coddington E. A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, Inc., New York (1955).
11. Cowan, C., Esposito, P., Ghoussoub, N., Moradifam, A.: The critical dimension for a fourth order elliptic problem with singular nonlinearity. Arch. Ration. Mech. Anal. (2009, to appear).
12. Dávila J., Dupaigne L., Guerra I., Montenegro M.: Stable solutions for the bilaplacian with exponential nonlinearity. SIAM J. Math. Anal. 39(2), 565-592 (2007).
13. Dávila, J., Flores, I., Guerra, I.: Multiplicity of solutions for a fourth order problem with exponential nonlinearity. J. Differ. Equ. (2009). doi: 10. 1016/j. jde. 2009. 07. 023.
14. Ferrero A., Grunau H.-C.: The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity. J. Differ. Equ. 234(2), 582-606 (2007).
15. Ferrero A., Grunau H.-C., Karageorgis P.: Supercritical biharmonic equations with power-type nonlinearity. Ann. Mat. Pura Appl. 188, 171-185 (2009).
16. Ferrero A., Warnault G.: On solutions of second and fourth order elliptic equations with power-type nonlinearities. Nonlinear Anal. 70(8), 2889-2902 (2009).
17. Flores I.: A resonance phenomenon for ground states of an elliptic equation of Emden-Fowler type. J. Differ. Equ. 198(1), 1-15 (2004).
18. Flores I.: Singular solutions of the Brezis-Nirenberg problem in a ball. Commun. Pure Appl. Anal. 8(2), 673-682 (2009).
19. Dolbeault J., Flores I.: Geometry of phase space and solutions of semilinear elliptic equations in a ball. Trans. Am. Math. Soc. 359(9), 4073-4087 (2007).
20. Gazzola F., Grunau H.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334(4), 905-936 (2006).
21. Gelfand I. M.: Some problems in the theory of quasilinear equations. Section 15, due to G. I. Barenblatt. Am. Math. Soc. Trans. II Ser. 29, 295-381 (1963).
22. Gilbarg D., Trudinger N. S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001).
23. Gidas B., Ni W. M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209-243 (1979).
24. Grunau H., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307(4), 589-626 (1997).
25. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Applied Mathematical Sciences, vol. 42. Springer, New York (1990).
26. Guo Z., Wei J.: On a fourth order nonlinear elliptic equation with negative exponent. SIAM J. Math. Anal. 40(5), 2034-2054 (2009).
27. 3. Adv. Differ. Equ. 13(7-8), 753-780 (2008).
28. Guo, Z., Wei, J.: Entire solutions and global bifurcations for a biharmonic equation with singular nonlinearity II (preprint).
29. Hartman, P.: Ordinary differential equations. Classics in Applied Mathematics, vol. 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002).
30. Joseph D. D., Lundgren T. S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241-269 (1972).
31. Karageorgis, P.: Stability and intersection properties of solutions to the nonlinear biharmonic equation (preprint).
32. Lin F., Yang Y.: Nonlinear non-local elliptic equation modelling electrostatic actuation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2081), 1323-1337 (2007).
33. Lions P. L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24(4), 441-467 (1982).
34. McKenna, P. J., Reichel, W.: Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry. Electron. J. Differ. Equ., no. 37, 13 pp. (2003).
35. Mitidieri, E., Pokhozhaev, S. I.: A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 3-383 (2001). English translation: Proc. Steklov Inst. Math. 234, 1-362 (2001).
36. Palis J.: On Morse-Smale dynamical systems. Topology 8, 385-404 (1968).
37. Pelesko J. A., Bernstein A. A.: Modeling MEMS and NEMS. Chapman Hall and CRC Press, Boca Raton (2002).
38. Rellich, F.: Halbbeschränkte Differentialoperatoren höherer Ordnung. In: Gerretsen, J. C. H., et al. (eds.) Proceedings of the international congress of mathematicians Amsterdam 1954, vol. III, pp. 243-250. Nordhoff, Groningen (1956).
39. Ruelle D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, Inc., Boston (1989).