×

Problems for elliptic singular equations with a gradient term. (English) Zbl 1103.35031

The author studies the Dirichlet problem \(\Delta u+g(u)| \nabla u| ^q +f(u)=0\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a bounded smooth domain in \(\mathbb R^n\), \(0<q<2\), \(g(t)\) is a decreasing smooth function in \((0,\infty )\) and \(f(t)\) is a decreasing and positive smooth function in \((0,\infty )\), which approaches infinity as \(t\to \infty \). The existence and uniqueness of a positive classical solution of the problem is proved. The asymptotic behaviour of a solution near the boundary is studied.

MSC:

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bandle, C.; Giarrusso, E., Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Adv. differential equations, 1, 133-150, (1996) · Zbl 0840.35034
[2] Bandle, C.; Marcus, M., On second order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Differential integr. equations, 11, 23-34, (1998) · Zbl 1042.35535
[3] S. Berhanu, F. Cuccu, G. Porru, On the boundary behaviour, including second order effects, of solutions to singular elliptic problems, Acta Math. Sin., Eng. Ser., in print. · Zbl 1163.35391
[4] Berhanu, S.; Gladiali, F.; Porru, G., Qualitative properties of solutions to elliptic singular problems, J. inequal. appl., 3, 313-330, (1999) · Zbl 0931.35019
[5] Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Commun. partial differential equations, 2, 193-222, (1997) · Zbl 0362.35031
[6] Giarrusso, E., Asymptotic behaviour of large solutions of an elliptic quasilinear equation in a borderline case, C.R. acad. sci. Paris Sér. I math., 331, 777-782, (2000) · Zbl 0966.35041
[7] Giarrusso, E.; Porru, G., Boundary behaviour of solutions to nonlinear elliptic singular problems, (), 163-178
[8] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer Berlin · Zbl 0691.35001
[9] Kawohl, B., On a class of singular elliptic equations, (), 156-163 · Zbl 0821.35053
[10] Kazdan, J.L.; Kramer, R.J., Invariant criteria for existence of solutions to quasilinear elliptic equations, Commun. pure appl. math., 31, 619-645, (1989) · Zbl 0368.35031
[11] Ladyzhenskaya, O.A.; Ural’tseva, N.N., Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[12] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. am. math. soc., 111, 721-730, (1991) · Zbl 0727.35057
[13] Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana univ. math. J., 21, 979-1000, (1972) · Zbl 0223.35038
[14] Stuart, C.A., Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147, 53-63, (1976) · Zbl 0324.35037
[15] Zhang, Z.; Cheng, J., Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear anal., 57, 473-484, (2004) · Zbl 1096.35050
[16] Zhang, Z.; Yu, J., On a singular nonlinear Dirichlet problem with a convection term, SIAM J. math. anal., 32, 916-927, (2000) · Zbl 0988.35059
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.