Publication: Branches of positive and free boundary solutions for some singular quasilinear elliptic problems
Loading...
Files
Full text at PDC
Publication Date
2009-04-01
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
We study the existence and multiplicity of solutions, strictly positive or nonnegative having a free boundary (the boundary of the set where the solution vanishes) of some one-dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms.
Description
UCM subjects
Unesco subjects
Keywords
Citation
S.N. Antontsev, J.I. Díaz, S.I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Birkhäuser-Verlag, Basel, 2001.
D. Aronson, M.G. Crandall, L.A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 16 (1982) 1001–1022.
C. Bandle, A. Pozio, A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 303 (1987) 487–501.
C. Bandle, R.P. Sperb, I. Stakgold, Diffusion and reaction with monotone kinetics, Nonlinear Anal. 8 (1984) 321–333.
C. Bandle, I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc. 286 (1984) 275–293.
Ph. Benilan, H. Brezis, M.G. Crandall, A semilinear equation in L1(RN), Ann. Sc. Norm. Super. Pisa 4 (2) (1975) 523–555.
A. Bensoussan, H. Brezis, A. Friedman, Estimates on the free boundary for quasi variational inequalities, Comm. Partial Differential Equations 2 (3) (1977) 297–321.
F. Bernis, Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n, Comm. Partial Differential Equations 9 (3) (1984) 271–312.
M.F. Bidaut-Véron, V. Galaktionov, P. Grillot, L. Véron, Singularities for a 2-dimensional semilinear elliptic equation with a non-Lipschitz nonlinearity, J. Differential Equations 154 (1999) 318–338.
H. Brezis, Solutions of variational inequalities with compact support, Uspekhi Mat. Nauk 129 (1974) 103–108.
C. Cortázar, M. Elgueta, P. Felmer, On a semilinear elliptic problem in RN with a non-Lipschitz nonlinearity, Adv. Differential Equations 1 (1996) 199–218.
Y.S. Choi, A.C. Lazer, P.J. McKenna, Some remarks on a singular elliptic boundary value problem, Nonlinear Anal. 32 (1998) 305–314.
F. Conrad, C.M. Brauner, F. Issard-Roch, B. Nicolaenko, Nonlinear eigenvalue problems in elliptic variational inequalities: A local study, Comm. Partial Differential Equations 10 (2) (1985) 151–190.
M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193–222
J. Dávila, M. Montenegro, A singular equation with positive and free boundary solutions, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 97 (2003) 107–112.
J. Dávila, M. Montenegro, Positive versus free boundary solutions to a singular equation, J. Anal. Math. 90 (2003) 303–335.
J. Dávila, M. Montenegro, Nonlinear problems with solutions exhibiting a free boundary on the boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 303–330.
J.I. Díaz, Técnica de supersoluciones locales para problemas estacionarios no lineales: aplicación al cálculo de flujos subsónicos, Memoria 14, Real Academia de Ciencias, 1980.
J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985.
J.I. Díaz, J. Hernández, On the existence of a free boundary for a class of reaction-diffusion systems, SIAM J. Math. Anal. 15 (1984) 670–685.
J.I. Díaz, J. Hernández, Qualitative properties of free boundaries for some nonlinear degenerate parabolic equations, in: L. Boccardo, A. Tesei (Eds.), Nonlinear Parabolic Equations: Qualitative Properties of Solutions, Longman, London, 1987, pp. 85–93.
J.I. Díaz, J. Hernández, Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris Ser. I 329 (1999) 587–592.
J.I. Díaz, M.A. Herrero, Proprietés de support compact pour certaines équations elliptiques et paraboliques non linéaires, C. R. Acad. Sci. Paris Ser. I 286 (1978) 815–817.
J.I. Díaz, M.A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 98 (1981) 249–258.
J.I. Díaz, J.M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987) 1333–1344.
J. Eisner, M. Kučera, L. Recke, Bifurcation direction and exchange of stability for variational inequalities on nonconvex sets, Nonlinear Anal. 67 (4) (2007) 1082–1101.
A. Friedman, D.S. Phillips, The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc. 282 (1984) 153–182.
W. Fulks, J.S. Maybee, A singular nonlinear equation, Osaka J. Math. 12 (1960) 1–19
M. Ghergu, V.D. Rǎdulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford University Press, Oxford, 2008.
R. Glowinski, J.L. Lions, R. Tremolières, Analyse Numérique des Inéquations Variationelles, vols. 1 & 2, Dunod, Paris, 1976.
M.E. Gurtin, R.C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977) 35–49
Y. Haitao, Positive versus compact support solutions to a singular elliptic problem, J. Math. Anal. Appl. 319 (2006) 830–840.
J. Hernández, Some free boundary problems for predator-prey systems with nonlinear diffusion, Proc. Sympos. Pure Math. 45 (1986) 481–488.
J. Hernández, J. Karátson, P.L. Simon, Multiplicity for semilinear elliptic equations involving singular nonlinearity, Nonlinear Anal. 65 (2006) 265–283.
J. Hernández, F.J. Mancebo, Singular elliptic and parabolic equations, in: M. Chipot, P. Quittner (Eds.), Handbook Differential Equations, vol. 3, Elsevier, Amsterdam, 2006, pp. 317–400.
J. Hernández, F.J. Mancebo, J.M. Vega, On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (6) (2002) 777–813.
J. Hernández, F.J. Mancebo, J.M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 41–62.
T. Horváth, P.L. Simon, On the exact number of solutions of a singular boundary value problem, preprint, 2008.
P. Huang, S. Wang, T. Yeh, Classification of bifurcation diagrams of a p-Laplacian semipositone problem, preprint, 2008.
S. Kamin, L. Véron, Flat core properties associated to the p-Laplace operator, Proc. Amer. Math. Soc. 118 (4) (1993) 1079–1085.
J. Karátson, P.L. Simon, Bifurcations of semilinear elliptic equations with convex nonlinearity, Electron. J. Differential Equations 43 (1999) 1–16.
S. Kichenassamy, J. Smoller, On the existence of radial solutions of quasi-linear elliptic quations, Nonlinearity 16 (3) (1990) 677–694.
D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
T. Ouyang, J. Shi, M. Yao, Exact multiplicity and bifurcation of solutions of a singular equation, preprint, 1996.
A. Pozio, A. Tesei, Support properties of solutions for a class of degenerate parabolic problems, Comm. Partial Differential Equations 12 (1987) 47–75.
P. Pucci, M. García-Huidobro, R. Manásevich, J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights Ann. Mat. Pura Appl. 185 (2006) S205—S243.
P. Pucci, J. Serrin, The Maximum Principle, Progr. Nonlinear Differential Equations Appl., vol. 75, Birkhäuser Boston, Boston, MA, 2007.
J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 138 (1998) 1389–1401.
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983.
J. Smoller, A.G. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981) 269–290.
C.A. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z. 147 (1976) 53–63.
C.A. Stuart, Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities, Math. Z. 163 (1978) 239–249
J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202.
S. Wang, A correction for a paper by J. Smoller and A. Wasserman, J. Differential Equations 77 (1989) 199–202.