Stable and singular solutions of the equation Δu = 1/u

Indiana University Mathematics Journal

Volumn 53, Issue 6, 2004, Pages 1681-1703

  Meadows, Alexander M ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Semilinear elliptic equations; Singular solutions; Stable solutions

Indexed keywords

EID: 12744278657     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (22)

1. CLAUDE-MICHEL BRAUNER and BASIL NICOLAENKO, On nonlinear eigenvalue problems which extend into free boundaries problems, Bifurcation and Nonlinear Eigenvalue Problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978), Lecture Notes in Math., vol. 782, Springer, Berlin, 1980, pp. 61-100. MR572251 (83e:35127)
2. LUIS CAFFARELLI, ROBERT HARDT, and LEON SIMON, Minimal surfaces with isolated singularities, Manuscripta Math. 48 (1984), 1-18. MR753722 (86h:53007)
3. EARL A. CODDINGTON and NORMAN LEVINSON, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR0069338 (16,1022b)
4. 1minimizers of a nondifferentiable functional, Commun. Contemp. Math. 6 (2004), 165-193. MR2048779
5. JUAN DÁVILA and MARCELO MONTENEGRO, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303-335. MR2001074
6. KLAUS DEIMLING, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985, ISBN 3-540-13928-1. MR787404 (86j:47001)
7. JESÚS ILDEFONSO DIAZ, JEAN-MICHEL MOREL, and LUC OSWALD, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333-1344. MR912208 (89m:35077)
8. ULRICH DIERKES, Singular minimal surfaces, Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003, pp. 177-193. MR2008338 (2004m:53012)
9. JESÚS P. GARCÍA AZORERO and IRENEO PERAL ALONSO, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441-476. MR1616905 (99f:35099)
10. BASILIS GIDAS, WEI MING Ni, and LOUIS NIRENBERG, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR544879 (80h:35043)
11. DAVID GILBARG and NEIL S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, ISBN 3-540-41160-7. MR1814364 (2001k:35004)
12. GODFREY HARDY, JOHN E. LITTLEWOOD, and GEORGE PÓLYA, inequalities, Cambridge Univ. Press, Cambridge, 1934.
13. CHARLES B. MORREY Jr., Multiple Integrals in The Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR0202511 (34 #2380)
14. DANIEL PHILLIPS, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1-17. MR684751 (84e:49012)
15. _, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), 1409-1454. MR714047 (85b:35068)
16. LEON SIMON, Some examples of singular minimal hypersurfaces (2001) (in preparation).
17. _, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983, ISBN 0-86784-429-9. MR756417 (87a:49001)
18. _, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), 281-326. MR1243525 (94k:58028)
19. _, The minimal surface equation, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 239-272. MR1490041 (99b:53014)
20. MICHAEL STRUWE, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 34, Springer-Verlag, Berlin, 1996, ISBN 3-540-58859-0. MR1411681 (98f:49002)
21. DONALD SARASON, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. MR0377518 (51 #13690)
22. 2 are Lipschitz manifolds, J. Differential Geom. 39 (1994), 65-101. MR1258915 (95b:49066)