Research and Innovation Articles
Published 2010-09-21
Keywords
- Singular equations,
- free boundary
How to Cite
Dávila, J., & Montenegro, M. (2010). Remarks on positive and free boundary solutions to a singular equation. Revista Integración, Temas De matemáticas, 28(2), 85–100. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2169
Abstract
The equation −∆u = χ{u>0} (− 1/(u^β) + λf(x, u) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥ 0 for every λ > 0. For λ less than a constant λ ∗ the solution vanishes inside the domain, and for λ > λ∗ the solution is positive and stable. We obtain optimal regularity of uλ even in the presence of the free boundary. If 0 < λ < λ∗ the solutions of the singular parabolic equation ut − ∆u + 1/(u^β) = λf(u) quench in finite time, and for λ > λ∗ the solutions are globally positively defined.
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References
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[16] Montenegro M. and Queiroz O., “Existence and regularity to an elliptic equation with logarithmic nonlinearity”, J. Differential Equations, 246 (2009), 482–511.
[17] Phillips D., “A minimization problem and the regularity of solutions in the presence of a free boundary”, Indiana Univ. Math. J., 32 (1983), 1–17.
[18] Shi J. and Yao M., “On a singular nonlinear semilinear elliptic problem”, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1389–1401.
[2] Brezis H., Cazenave T., Martel Y., and Ramiandrisoa A., “Blow-up for ut − ∆u = g(u) revisited”, Adv. Differential Equations, 1 (1996), 73–90.
[3] Brezis H. and Marcus M., “Hardy’s inequalities revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1998), 217–237.
4] Choi Y.S., Lazer A.C., and McKenna P.J., “Some remarks on a singular elliptic boundary value problem”, Nonlinear Anal., 32 (1998), 305–314.
[5] Crandall M.G., Rabinowitz P.H and Tartar L., “On a Dirichlet problem with a singular nonlinearity”, Comm. Partial Differential Equations, 2 (1977), 193–222.
[6] Dávila J. and Montenegro M., “Radial solutions of an elliptic equation with singular nonlinearity”, J. Math. Anal. Appl., 352 (2009), 360–379.
[7] Dávila J., “Global regularity for a singular equation and local H1 minimizers of a nondifferentiable
functional”, Commun. Contemp. Math., 6 (2004), 165–193.
[8] Dávila J. and Montenegro M., “Existence and asymptotic behavior for a singular parabolic equation”, Trans. Amer. Math. Soc., 357 (2005), 1801–1828.
[9] Dávila J. and Montenegro M., “Concentration for an elliptic equation with singular nonlinearity”, to appear in J. Math. Pures Appl., (2011), doi:10.1016/j.matpur.2011.02.001
[10] Díaz J.I., Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Research Notes in Mathematics, 106. Pitman (Advanced Publishing Program), Boston, MA, 1985.
[11] Díaz J.I., Morel J.M., and Oswald L., “An elliptic equation with singular nonlinearity”, Comm. Partial Differential Equations, 12 (1987), 1333–1344.
[12] Giaquinta M. and Giusti E., “Sharp estimates for the derivatives of local minima of variational integrals”, Boll. Un. Mat. Ital. A, 3 (1984), 239–248.
[13] Gui C. and Lin F.H., “Regularity of an elliptic problem with a singular nonlinearity”, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 1021–1029.
[14] Martel Y., “Uniqueness of weak extremal solutions of nonlinear elliptic problems”, Houston J. Math., 23 (1997), 161–168.
[15] Mignot F. and Puel J.P., “Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe”, Comm. Partial Differential Equations, 5 (1980), 791–836.
[16] Montenegro M. and Queiroz O., “Existence and regularity to an elliptic equation with logarithmic nonlinearity”, J. Differential Equations, 246 (2009), 482–511.
[17] Phillips D., “A minimization problem and the regularity of solutions in the presence of a free boundary”, Indiana Univ. Math. J., 32 (1983), 1–17.
[18] Shi J. and Yao M., “On a singular nonlinear semilinear elliptic problem”, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1389–1401.