Stable solutions of elliptic partial differential equations

Stable Solutions of Elliptic Partial Differential Equations

Volumn , Issue , 2011, Pages 1-317

  Dupaigne, Louis ^a  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ELLIPTIC EQUATIONS; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; FUNCTIONAL INEQUALITIES; HIGHER ORDER EQUATIONS; INVERSE SQUARE POTENTIAL; MATHEMATICAL PHYSICS; REGULARITY THEORIES; TRANSITION THEORIES;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 85059280742     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Book

References (225)

1. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727.
2. A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Math. Acad. Sci. Paris 339 (2004), no. 5, 339-344 (English, with English and French summaries).
3. G. Alberti, L. Ambrosio, and X. Cabré, On a long-standing conjecture of E De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9-33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.
4. N. D. Alikakos and P. W. Bates, On the singular limit in a phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 141-178 (English, with French summary).
5. W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491.
6. F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. (2) 84 (1966), 277-292.
7. L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725-739 (electronic).
8. L. Ambrosio, G. De Philippis, and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Preprint available at http://cvgmt.sns.it/cgi/get.cgi/people/dephilipp/ (2010).
9. G. Arioli, F. Gazzola, and H.-C. Grunau, Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity, J. Differential Equations 230 (2006), no. 2, 743-770, DOI 10.1016/j.jde.2006.05.015.
10. G. Arioli, F. Gazzola, H.-C. Grunau, and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (2005), no. 4, 1226- 1258 (electronic), DOI 10.1137/S0036141002418534.
11. S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001.
12. S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations 6 (1998), no. 1, 1-38.
13. J. Bebernes and D. Eberly, Mathematical problems from combustion theory, Applied Mathematical Sciences, vol. 83, Springer-Verlag, New York, 1989.
14. H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SIAM J. Math. Anal. 18 (1987), no. 3, 722-727, DOI 10.1137/0518055.
15. E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differential Equations (2005), No. 34, 20 pp. (electronic).
16. H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Symmetry for elliptic equations in a half space, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 27-42.
17. H. Berestycki, L. A. Caffarelli, and L. Nirenberg Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69-94 (1998). Dedicated to Ennio De Giorgi.
18. H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1-37.
19. S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), no. 1, 551-558, DOI 10.1007/BF01475472 (German).
20. M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489- 539, DOI 10.1007/BF01243922.
21. T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo (1905), 97-135.
22. E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243-268.
23. D. Bonheure, V. Bouchez, C. Grumiau, and J. Van Schaftingen, Asymptotics and symmetries of least energy nodal solutions of Lane-Emden problems with slow growth, Commun. Contemp. Math. 10 (2008), no. 4, 609-631, DOI 10.1142/S0219199708002910.
24. V. Bouchez, Nonlinear boundary value problems: existence, estimates, symmetry, PhD Thesis, Université catholique de Louvain 167 (2009).
25. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011 (English).
26. H. Brezis, Is there failure of the inverse function theorem?, Morse theory, minimax theory and their applications to nonlinear differential equations, New Stud. Adv. Math., vol. 1, International Press, Somerville, MA, 2003, pp. 23-33.
27. H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 2, 223-262 (English, with Italian summary).
28. H. Brezis and T. Cazenave, Nonlinear heat equation, unpublished (English).
29. t-&u = g(u) revisited, Adv. Differential Equations 1 (1996), no. 1, 73-90.
30. H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of -u = V(x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223-1253.
31. H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477, DOI 10.1002/cpa.3160360405.
32. H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443-469.
33. C. C. Burch, The Dini condition and regularity of weak solutions of elliptic equations, J. Differential Equations 30 (1978), no. 3, 308-323, DOI 10.1016/0022-0396(78)90003- 7.
34. X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, arXiv:0909.4696v1.
35. X. Cabré, Regularity of radial extremal solutions of semilinear elliptic equations, Bol. Soc. Esp. Mat. Apl. S~eMA (2006), no. 34, 92-98.
36. X. Cabré, Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations, Discrete Contin. Dyn. Syst. 8 (2002), no. 2, 331-359. Current developments in partial differential equations (Temuco, 1999).
37. X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.) 4 (1998), no. 1, 1-10, DOI 10.1007/s000290050022.
38. X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), no. 2, 709-733.
39. X. Cabré and A. Capella, Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations, Pure Appl. Math. Q. 3 (2007), no. 3, 801-825.
40. X. Cabré, A. Capella, and M. Sanchón, Regularity of radial minimizers of reaction equations involving the p-Laplacian, Calc. Var. Partial Differential Equations 34 (2009), no. 4, 475-494, DOI 10.1007/s00526-008-0192-3.
41. X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, arXiv:1004.2866v1.
42. X. Cabré and M. Sanchón, Semi-stable and extremal solutions of reaction equations involving the p-Laplacian, Commun. Pure Appl. Anal. 6 (2007), no. 1, 43-67.
43. X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), no. 12, 1678-1732, DOI 10.1002/cpa.20093.
44. L. Caffarelli, R. Hardt, and L. Simon, Minimal surfaces with isolated singularities, Manuscripta Math. 48 (1984), no. 1-3, 1-18.
45. L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111-1144.
46. L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, posted on 2010, DOI 10.1007/s00526-010-0359-6, (to appear in print).
47. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260, DOI 10.1080/03605300600987306.
48. A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. (1952), 85-139.
49. S. Campanato, Equazioni ellittiche del II deg ordine espazi L(2,λ), Ann. Mat. Pura Appl. (4) 69 (1965), 321-381 (Italian).
50. A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear elliptic equations, arXiv:1004.1906v1.
51. H. Cartan, Differential calculus, Hermann, Paris, 1971. Exercises by C. Buttin, F. Rideau and J. L. Verley; Translated from the French.
52. D. Castorina, P. Esposito, and B. Sciunzi, Degenerate elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations 34 (2009), no. 3, 279-306, DOI 10.1007/s00526-008-0184-3.
53. D. Castorina, P. Esposito, and B. Sciunzi, Low dimensional instability for semilinear and quasilinear problems in RN, Commun. Pure Appl. Anal. 8 (2009), no. 6, 1779-1793, DOI 10.3934/cpaa.2009.8.1779.
54. A. Castro, J. Cossio, and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), no. 4, 1041-1053, DOI 10.1216/rmjm/1181071858.
55. D. Cassani, J. M. do Ó, and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud. 9 (2009), no. 1, 177-197.
56. T. Cazenave, M. Escobedo, and M. A. Pozio, Some stability properties for minimal solutions of -u = λg(u), Port. Math. (N.S.) 59 (2002), no. 4, 373-391.
57. W. X. Chen and C. Li, A priori estimates for solutions to nonlinear elliptic equations, Arch. Rational Mech. Anal. 122 (1993), no. 2, 145-157, DOI 10.1007/BF00378165.
58. W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), no. 3, 615-622.
59. P. Clément, D. G. de Figueiredo, and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 133-170.
60. C. Cowan, P. Esposito, and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general doamins, DCDS 28 (2010), no. 3.
61. C. Cowan, P. Esposito, N. Ghoussoub, and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Rat. Mech. Anal. (2010).
62. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, 1953.
63. M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. (1975), no. 3, 207-218.
64. M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. Math. (2) 106 (1977), no. 1, 93-100.
65. R. Dalmasso, Symmetry properties in higher-order semilinear elliptic equations, Nonlinear Anal. 24 (1995), no. 1, 1-7.
66. L. Damascelli, A. Farina, B. Sciunzi, and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, 1099-1119, DOI 10.1016/j.anihpc.2008.06.001.
67. L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 (2004), no. 2, 483-515, DOI 10.1016/j.jde.2004.05.012.
68. E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000), 225-233, DOI 10.1007/BF02505896.
69. E. N. Dancer, Weakly nonlinear Dirichlet problems on long or thin domains, Mem. Amer. Math. Soc. 105 (1993), no. 501, viii+66.
70. E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), no. 3, 425-434, DOI 10.1017/S0004972700012089.
71. E. N. Dancer and A. Farina, On the classification of solutions of u = eu on RN: stability outside a compact set and applications, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1333- 1338, DOI 10.1090/S0002-9939-08-09772-4.
72. J. Dávila, M. del Pino, M. Musso, and J. Wei, Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var. Partial Differential Equations 32 (2008), no. 4, 453-480, DOI 10.1007/s00526-007-0154-1.
73. J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math. 9 (2007), no. 5, 639-680.
74. J. Dávila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 335-365.
75. J. Dávila and L. Dupaigne, Comparison results for PDEs with a singular potential, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 1, 61-83.
76. J. Dávila, L. Dupaigne, and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, to appear in J. Functional Analysis.
77. J. Dávila, L. Dupaigne, I. Guerra, and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal. 39 (2007), no. 2, 565-592 (electronic), DOI 10.1137/060665579.
78. J. Dávila, L. Dupaigne, and M. Montenegro, The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal. 7 (2008), no. 4, 795-817, DOI 10.3934/cpaa.2008.7.795.
79. J. Dávila, I. Flores, and I. Guerra, Multiplicity of solutions for a fourth order problem with exponential nonlinearity, J. Differential Equations 247 (2009), no. 11, 3136-3162, DOI 10.1016/j.jde.2009.07.023.
80. J. Dávila, I. Flores, and I. Guerra, Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Annalen 348 (2010), no. 1, 143-193.
81. E. De Giorgi, Convergence problems for functionals and operators, Analysis (Rome, 1978), Pitagora, Bologna, 1979, pp. 131-188.
82. E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 79-85 (Italian).
83. D. G. de Figueiredo, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41-63.
84. M. del Pino, M. Kowalczyk, and J. Wei, A counterexample to a conjecture by De Giorgi in large dimensions, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1261-1266 (English, with English and French summaries).
85. E. DiBenedetto, C1+ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827-850, DOI 10.1016/0362-546X(83)90061-5.
86. W. Ding, personal communication to W. X. Chen and C. Li.
87. J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Henri Poincaré 10 (2010), no. 7, 1311-1333, DOI 10.1007/s00023-009-0016-9.
88. L. Dupaigne and A. Farina, Stable solutions of u = f (u) in RN, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 855-882, DOI 10.4171/JEMS/217.
89. L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal. 70 (2009), no. 8, 2882-2888, DOI 10.1016/j.na.2008.12.017.
90. L. Dupaigne and Y. Sire, A Liouville theorem for non local elliptic equations, arXiv:0909.1650v1.
91. P. Esposito, N. Ghoussoub, and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, vol. 20, Courant Institute of Mathematical Sciences, New York, 2010.
92. M. J. Esteban and P.-L. Lions, Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 1- 2, 1-14.
93. L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998.
94. A. Farina, Liouville-type results for solutions of -u = jujp1u on unbounded domains of RN, C. R. Math. Acad. Sci. Paris 341 (2005), no. 7, 415-418 (English, with English and French summaries).
95. A. Farina, Stable solutions of -u = eu on RN, C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 63-66 (English, with English and French summaries).
96. A. Farina, Liouville-type theorems for elliptic problems, Handbook of Differential Equations: Stationary Partial Differential Equations., vol. 4, Edited by M.Chipot, Elsevier B.V., 2007.
97. A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of RN, J. Math. Pures Appl. (9) 87 (2007), no. 5, 537-561 (English, with English and French summaries).
98. A. Farina, One-dimensional symmetry for solutions of quasilinear equations in R2, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6 (2003), no. 3, 685-692 (English, with English and Italian summaries).
99. A. Farina, B. Sciunzi, and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 741-791.
100. A. Farina, B. Sciunzi, and E. Valdinoci, On a Poincaré type formula for solutions of singular and degenerate elliptic equations, Manuscripta Mathematica 132 (2010), no. 3-4, 335-342.
101. A. Farina, Y. Sire, and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, arxiv:0809.3025v1.
102. A. Farina and E. Valdinoci, Flattening Results for Elliptic PDEs in Unbounded Domains with Applications to Overdetermined Problems, Archive for Rational Mechanics and Analysis 195 (2010), no. 3, 1025-1058.
103. A. Farina and E. Valdinoci, Rigidity results for elliptic PDES with uniform limits: an abstract framework with applications, to appear in Indiana Univ. Math. J.
104. A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, Recent progress on reaction-diffusion systems and viscosity solutions, World Scientific Publishing, Hackensack, NJ, 2009, pp. 74-96.
105. A. Ferrero, H.-C. Grunau, and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl. (4) 188 (2009), no. 1, 171-185, DOI 10.1007/s10231-008-0070-9.
106. A. Ferrero and G. Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal. 70 (2009), no. 8, 2889-2902, DOI 10.1016/j.na.2008.12.041.
107. M. Fila and H. Matano, Connecting equilibria by blow-up solutions, Discrete Contin. Dynam. Systems 6 (2000), no. 1, 155-164.
108. M. Fila and P. Polácik, Global solutions of a semilinear parabolic equation, Adv. Differential Equations 4 (1999), no. 2, 163-196.
109. D. L. Finn, Convexity of level curves for solutions to semilinear elliptic equations, Commun. Pure Appl. Anal. 7 (2008), no. 6, 1335-1343, DOI 10.3934/cpaa.2008.7.1335.
110. D. A. Frank-Kamenetskii, Diffusion and heat exchange in chemical kinetics, Princeton University Press, Princeton, NJ, 1955.
111. H. Fujita, On the nonlinear equations u + eu = 0 and @ v=@ t = v + ev, Bull. Amer. Math. Soc. 75 (1969), 132-135.
112. H. Fujita, On the asymptotic stability of solutions of the equation vt = v + ev, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 252-259.
113. J. García Azorero and I. Peral Alonso, On an Emden-Fowler type equation, Nonlinear Anal. 18 (1992), no. 11, 1085-1097, DOI 10.1016/0362-546X(92)90197-M.
114. J. García Azorero, I. Peral Alonso, and J.-P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal. 22 (1994), no. 4, 481-498, DOI 10.1016/0362-546X(94)90169-4.
115. C. F. Gauss, Werke: herausgegeben von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Georg Olms Verlag, New York (1863).
116. F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006), no. 4, 905-936, DOI 10.1007/s00208-005-0748-x.
117. F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic boundary value problems (Springer Berlin, ed.), Lecture notes in Mathematics, vol. 1991, 2010.
118. F. Gazzola and A. Malchiodi, Some remarks on the equation -u = λ(1+u)p for varying λ, p and varying domains, Comm. Partial Differential Equations 27 (2002), no. 3-4, 809-845, DOI 10.1081/PDE-120002875.
119. I. M. Gelofand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295-381.
120. N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481-491.
121. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Stud., vol. 105, Princeton University Press, Princeton, NJ, 1983.
122. M. Giaquinta and S. Hildebrandt, Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 310, Springer-Verlag, Berlin, 1996. The Lagrangian formalism.
123. B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243.
124. B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901, DOI 10.1080/03605308108820196.
125. B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525-598, DOI 10.1002/cpa.3160340406.
126. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
127. E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984.
128. Z. Guo and J. Li, The blow up locus of semilinear elliptic equations with supercritical exponents, Calc. Var. Partial Differential Equations 15 (2002), no. 2, 133-153, DOI 10.1007/s00526-002-0078-8.
129. E. Hopf, A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc. 3 (1952), 791-793.
130. E. Hopf, Elementare Bemarkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsber. d. Preuss. Akad. d. Wiss. 19 (1927), 147-152.
131. J. Jacobsen and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations 184 (2002), no. 1, 283-298.
132. D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241-269.
133. T. Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135-148 (1973).
134. B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985.
135. B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations 10 (1985), no. 10, 1213-1225, DOI 10.1080/03605308508820404.
136. J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Differential Equations 16 (1974), 103-125.
137. H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361-1376.
138. N. J. Korevaar, Convexity properties of solutions to elliptic PDEs, Variational methods for free surface interfaces (Menlo Park, Calif., 1985), Springer, New York, 1987, pp. 115- 121.
139. T. Laetsch, On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl. 35 (1971), 389-404.
140. A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), no. 6, 1350-1366, DOI 10.1137/0143090.
141. P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309-318.
142. E. Lieb, Bounds on the eigenvalues of the Laplace and Schroedinger operators, Bull. Amer. Math. Soc. 82 (1976), no. 5, 751-753.
143. G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203-1219, DOI 10.1016/0362-546X(88)90053-3.
144. C. S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc. 102 (1988), no. 2, 271-277, DOI 10.2307/2045874.
145. J.-L. Lions, Théorèmes de trace et d'interpolation. I, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 389-403.
146. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.
147. J. Liouville, Sur l'équation aux différences partielles d2 logλ dudv - λ 2a2=0., J. Math. Pures Appl. (1853), 71-72.
148. H. Lou, On singular sets of local solutions to p-Laplace equations, Chin. Ann. Math. Ser. B 29 (2008), no. 5, 521-530, DOI 10.1007/s11401-007-0312-y.
149. Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math. 23 (1997), no. 1, 161-168.
150. t-&u = g(u), Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 6, 687-723 (English, with English and French summaries).
151. A. Mellet and J. Vovelle, Existence and regularity of extremal solutions for a meancurvature equation, J. Differential Equations 249 (2010), no. 1, 37-75.
152. P. Mironescu and V. D. Radulescu, The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal. 26 (1996), no. 4, 857-875.
153. J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of Rn, Comm. Pure Appl. Math. 26 (1973), 361-379.
154. F. Mignot and J.-P. Puel, Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), no. 8, 791- 836 (French).
155. F. Mignot and J.-P. Puel, Quelques résultats sur un problème elliptique avec non linéarité exponentielle, Équations aux dérivées partielles et applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998, pp. 683-704 (French, with French summary).
156. L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123-142.
157. L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 843- 850.
158. L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 679-684.
159. M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2429-2438.
160. A. Moradifam, The singular extremal solutions of the bi-Laplacian with exponential nonlinearity, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1287-1293, DOI 10.1090/S0002- 9939-09-10257-5.
161. A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations 248 (2010), no. 3, 594-616, DOI 10.1016/j.jde.2009.09.011.
162. C. G. Moreira and M. A. S. Ruas, The curve selection lemma and the Morse-Sard theorem, Manuscripta Math. 129 (2009), no. 3, 401-408, DOI 10.1007/s00229-009-0275-2.
163. J.-M. Morel and L. Oswald, A uniform formulation for Hopf maximum principle, unpublished preprint (1985).
164. A. P. Morse, The behavior of a function on its critical set, Ann. of Math. (2) 40 (1939), no. 1, 62-70.
165. K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation -u = λeu on circular domains, Math. Ann. 299 (1994), no. 1, 1-15, DOI 10.1007/BF01459770.
166. G. Nedev, Regularity of the extremal solution to semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 11, 997-1002 (English, with English and French summaries).
167. G. Nedev, Extremal solutions of semilinear elliptic equations, Unpublished notes.
168. M. Nicolesco, Sur les fonctions de n variables, harmoniques d'ordre p, Bull. Soc. Math. France 60 (1932), 129-151 (French).
169. O. A. Oleinik, On properties of solutions of certain boundary problems for equations of elliptic type, Mat. Sbornik N.S. 30(72) (1952), 695-702 (Russian).
170. R. Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications Inc., New York, 1986.
171. R. Osserman, Minimal varieties, Bull. Amer. Math. Soc. 75 (1969), 1092-1120.
172. F. Pacard, A note on the regularity of weak solutions of -u = u in Rn, n λ 3, Houston J. Math. 18 (1992), no. 4, 621-632.
173. F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math. 79 (1993), no. 2, 161-172, DOI 10.1007/BF02568335.
174. F. Pacard, Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: the supercritical case, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 5, 537-551 (English, with English and French summaries).
175. F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities, J. Funct. Anal. 192 (2002), no. 1, 271-282, DOI 10.1006/jfan.2001.3901.
176. F. Pacella and M. Ramaswamy, Symmetry of solutions of elliptic equations via maximum principles (M. Chipot, ed.), Handbook of differential equations: stationary partial differential equations. Vol. VI, Elsevier/North-Holland, Amsterdam, 2008.
177. F. Pacella and T. Weth, Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1753-1762 (electronic), DOI 10.1090/S0002-9939-07-08652-2.
178. J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.
179. M. Picone, Maggiorazione degli integrali delle equazioni totalmente paraboliche alle derivate parziali del secondo ordine, Ann. Mat. Pura Appl. 7 (1929), no. 1, 145-192 (Italian).
180. M. Picone, Maggiorazioni degli integrali di equazioni lineari ellittico-paraboliche alle derivate parziali del secondo ordine, Rend. Accad. Naz. Lincei. 5 (1927), no. 6, 138- 143 (Italian).
181. S. I. Pohožaev, On the eigenfunctions of the equation u+λf (u) = 0, Dokl. Akad. Nauk SSSR 165 (1965), 36-39 (Russian).
182. M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.
183. P. Pucci and J. Serrin, The maximum principle, Progress in Nonlinear Differential Equations and Their Applications, 73, Birkhäuser Verlag, Basel, 2007.
184. P. Quittner and P. Souplet, Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.
185. A. Ramiandrisoa, Blow-up for two nonlinear problems, Nonlinear Anal. 41 (2000), no. 7- 8, Ser. A: Theory Methods, 825-854, DOI 10.1016/S0362-546X(98)00312-5.
186. Y. Rébaï, Solutions of semilinear elliptic equations with one isolated singularity, Differential Integral Equations 12 (1999), no. 4, 563-581.
187. G. V. Rozenbljum, Distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972), 1012-1015 (Russian).
188. M. V. Safonov, Boundary estimates for positive solutions to second order elliptic equations, http://arxiv.org/abs/0810.0522v2.
189. L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002.
190. M. Sanchón, Regularity of the extremal solution of some nonlinear elliptic problems involving the p-Laplacian, Potential Anal. 27 (2007), no. 3, 217-224, DOI 10.1007/s11118-007-9053-5.
191. M. Sanchón, Boundedness of the extremal solution of some p-Laplacian problems, Nonlinear Anal. 67 (2007), no. 1, 281-294, DOI 10.1016/j.na.2006.05.010.
192. A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. (1942), 883-890.
193. D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979-1000.
194. O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41-78.
195. O. Savin and E. Valdinoci, convergence for nonlocal phase transitions, arXiv:1007.1725v1.
196. O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, arXiv:1007.2114v1.
197. R. Schaaf, Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry, Adv. Differential Equations 5 (2000), no. 10-12, 1201-1220.
198. J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. (1934), no. 1, 257-282, DOI 10.1007/BF01170635 (German).
199. K. Schmitt, Positive solutions of semilinear elliptic boundary value problems, Topological methods in differential equations and inclusions (Montreal, PQ, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472, Kluwer Academic Publishers, Dordrecht, 1995, pp. 447-500.
200. R. T. Seeley, Spherical harmonics, Amer. Math. Monthly 73 (1966), no. 4, 115-121.
201. J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302.
202. H. S. Shapiro and M. Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36 (1994), no. 1, 99-101, DOI 10.1137/1036005.
203. L. Simon, The minimal surface equation, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 239-272.
204. J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105.
205. Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), no. 6, 1842- 1864, DOI 10.1016/j.jfa.2009.01.020.
206. G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965), Les Presses de l'Université de Montréal, Montreal, Que., 1966 (French).
207. P. Sternberg and K. Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199- 220.
208. P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63-85.
209. L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin, 2007.
210. J. F. Toland, On positive solutions of -u = F(x, u), Math. Z. 182 (1983), no. 3, 351- 357.
211. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150, DOI 10.1016/0022-0396(84)90105-0.
212. N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747.
213. E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S~eMA (2009), no. 49, 33-44.
214. J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), no. 1, 103- 153.
215. S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in RN, J. Math. Pures Appl. (9) 88 (2007), no. 3, 241-250 (English, with English and French summaries).
216. S. Villegas, Sharp estimates for semi-stable radial solutions of semilinear elliptic equations, http://fr.arxiv.org/abs/0906.1443v1.
217. G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity, Commun. Pure Appl. Anal. 8 (2009), no. 5, 1709-1723, DOI 10.3934/cpaa.2009.8.1709.
218. G. Warnault, Liouville theorems for stable radial solutions for the biharmonic operator, Asymptotic Analysis (2010).
219. K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, preprint.
220. K. Wang, Partial regularity of stable solutions to Emden equation, preprint.
221. X.-J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B 27 (2006), no. 6, 637-642, DOI 10.1007/s11401-006-0142-3.
222. J. Wei and X. Xu, Classification of solutions of higher-order conformally invariant equations, Math. Ann. 313 (1999), no. 2, 207-228, DOI 10.1007/s002080050258.
223. X.-F. Yang, Nodal sets and Morse indices of solutions of super-linear elliptic PDEs, J. Funct. Anal. 160 (1998), no. 1, 223-253, DOI 10.1006/jfan.1998.3301.
224. S. Zaremba, Sur un problème mixte relatif à l'équation de Laplace, Bull. Intern. de l' Acad. Sci. de Cracovie. Série A, Sci. Math. (1910), 313-344.
225. Z. X. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), no. 2, 309-334.