Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

Communications on Pure and Applied Mathematics

Volumn 60, Issue 12, 2007, Pages 1731-1768

  Esposito, Pierpaolo ^a   Ghoussoub, Nassif ^b   Guo, Yujin ^b  

^a ROMA TRE UNIVERSITY   (Italy)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36248990914     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20189     Document Type: Article

References (20)

1. Alama, S.; Tarantello, G. Elliptic problems with nonlinearities indefinite in sign. J. Funct. Anal. 141 (1996), no. 1, 159-215.
2. Ambrosetti, A.; Brezis, H.; Cerami, G. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 ( 1994), no. 2, 519-543.
3. Ambrosetti, A.; Rabinowitz, P. H. Dual variational methods in critical point theory and application. J. Funct. Anal. 14 (1973), 349-381.
4. t - Δuw = g(u) revisited. Adv. Differential Equations 1 (1996), 73-90.
5. 1 local minimizers. C. R. Math. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465-472.
6. Brezis, H.; Vazquez, J. L. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Compl. Madrid 10 (1997), 443-469.
7. Cabré, X. Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems. Preprint, 2005. Availabe online at: http://www-mal.upc.es/~cabre/extr-subm.pdf.
8. n. C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 769-774.
9. Crandall, M. G.; Rabinowitz, P. H. Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58 (1975), 207-218.
10. Flores, G.; Mercado, G. A.; Pelesko, J. A. Dynamics and touchdown in electrostatic MEMS. Proceedings of the International Conference on MEMS, NANO, and Smart Systems (ICMENS'03), 182-187. Los Alamitos, Calif., IEEE Computer Society, 2003.
11. Ghoussoub, N.; Guo, Y. On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 38 (2007), 1423-1449.
12. Ghoussoub, N.; Guo, Y. On the partial differential equations of electrostatic MEMS devices II: dynamic case. NoDEA Nonlinear Differential Equations Appl., in press.
13. Guo, Y; Pan, Z.; Ward, M. J. Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J. Appl. Math. 166 (2005), no. 1, 309-338 (electronic).
14. Joseph, D. D.; Lundgren, T. S. Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49 (1973), 241-268.
15. Keener, J. P.; Keller, H. B. Positive solutions of convex nonlinear eigenvalue problems. J. Differential Equations 16 (1974), 103-125.
16. Martel, Y. Uniqueness of weak extremal solutions of nonlinear elliptic problems. Houston J. Math. 23 (1997), no. 1, 161-168.
17. Mignot, F.; Puel, J. P. Sur une classe de problèmes non linéaires avec non linéarité positive, croissante, convexe. Comm. Partial Differential Equations 5 (1980), no. 8, 791-836.
18. Pelesko, J. A. Mathematical modeling of electrostatic MEMS with tailored dielectric properties. SIAM J. Appl. Math. 62 (2001/2002), no. 3, 888-908 (electronic).
19. Pelesko, J. A.; Bernstein, D. H. Modeling MEMS and NEMS. Chapman and Hall/CRC, Boca Raton, Fla., 2003.
20. Zheng, G. New results on the formation of singularities for parabolic problems. Preprint, 2005.