zbMATH — the first resource for mathematics

Exact boundary behavior of the unique positive solution to some singular elliptic problems. (English) Zbl 1281.31006
Summary: We give an exact asymptotic of the unique solution to the following singular boundary value problem \(-\varDelta u =a(x)g(u)\), \(x \in \varOmega\), \(u > 0\), in \(\varOmega\), \(u|_{\partial \varOmega} = 0\). Here \(\varOmega\) is a \(C^2\)-bounded domain in \(\mathbb R^n\) \((n \geq 2)\), \(g \in C^1\big((0,\infty), (0,\infty)\big)\) is nonincreasing on \((0,\infty)\) with
\[ \lim_{t\rightarrow 0} g'(t)\int^t_0\frac{ds}{g(s)} = -C_g \leq 0, \]
and the function \(a\) is in \(C^\alpha_{\text{loc}} (\varOmega)\), \(0 < \alpha < 1\), satisfying
\[ 0 < a_1 = \liminf_{ d(x)\rightarrow 0}\frac{a(x)}{ h(d(x))} \leq\limsup_{d(x)\rightarrow 0}\frac{a(x)} {h(d(x))} = a_2 < \infty, \]
where \[ h(t) = c t^{-\lambda} \exp(\int^\eta_t\frac{z(s)}s ds), \] \(\lambda \leq 2\), \(c > 0\), and \(z\) is continuous on \([0, \eta]\) for some \(\eta > 0\) such that \(z(0) = 0\). Two applications of this result are also given. The first concerns the boundary behavior of the unique solution of \(-\varDelta u +\frac\beta u |\nabla u|^2 = a(x)g(u)\) that vanishes on the boundary, and the second concerns the behavior of \(u\) in the case where the open set \(\varOmega\) is annular and the behaviors of the function \(a\) on the interior boundary and the exterior boundary may be different.

31C15 Potentials and capacities on other spaces
34B27 Green’s functions for ordinary differential equations
35K10 Second-order parabolic equations
Full Text: DOI
[1] Crandall, M. G.; Rabinowitz, P. H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial. Differential Equations, 2, 193-222, (1997) · Zbl 0362.35031
[2] Fulks, J. S.; Maybee, J. S., A singular nonlinear elliptic equation, Osaka J. Math., 12, 1-19, (1960) · Zbl 0097.30202
[3] Ghergu, M.; Rădulescu, V. D., Singular elliptic problems: bifurcation and asymptotic analysis, (2008), Oxford University Press · Zbl 1159.35030
[4] Nachman, A.; Callegari, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38, 275-281, (1980) · Zbl 0453.76002
[5] Lazer, A. C.; Mckenna, P. J., On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111, 721-730, (1991) · Zbl 0727.35057
[6] Ghergu, M.; Rădulescu, V. D., Bifurcation and asymptotics for the Lane-Fowler equations, C.R. Math. Acad. Sci. Paris, 337, 259-264, (2003) · Zbl 1073.35087
[7] Zhang, Z., The asymptotic behaviour of the unique solution for the singular laneemdenfowler equation, J. Math. Anal. Appl., 332, 33-43, (2005)
[8] Ben Othman, S.; Mâagli, H.; Masmoudi, S.; Zribi, M., Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems, Nonlinear Anal., 71, 4137-4150, (2009) · Zbl 1177.35091
[9] Gontara, S.; Mâagli, H.; Masmoudi, S.; Turki, S., Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl., 369, 719-729, (2010) · Zbl 1196.35109
[10] Zhang, Z.; Li, Bo, The boundary behavior of the unique solution to a singular Dirichlet problem, J. Math. Anal. Appl., 391, 278-290, (2012) · Zbl 1241.35086
[11] Cîrstea, F.; Radulescu, V. D., Uniqueness of the blow-up boundary solution of logistic equations with absorbtion, C.R. Math. Acad. Sci. Paris, 335, 447-452, (2002) · Zbl 1183.35124
[12] Mohammed, A., Boundary asymptotic and uniqueness of solutions to the \(p\)-Laplacian with infinite boundary value, J. Math. Anal. Appl., 325, 480-489, (2007) · Zbl 1142.35412
[13] Maric, V., (Regular Variation and Differential Equations, Lecture Notes in Math., vol. 1726, (2000), Springer-Verlag Berlin) · Zbl 0946.34001
[14] Seneta, R., (Regular Varying Functions, Lectures Notes in Math., vol. 508, (1976), Springer-Verlag Berlin)
[15] Chemmam, R.; Mâagli, H.; Masmoudi, S.; Zribi, M., Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal., 1, 301-318, (2012) · Zbl 1277.31016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.