×

Sublinear singular elliptic problems with two parameters. (English) Zbl 1039.35042

Let \(\Omega\) be a smooth bounded domain in \(\mathbb R^N\) \((N\geq 2)\). The authors study the existence or nonexistence of solutions to the following boundary value problem \[ \begin{cases} -\Delta u+ K(x)g(u)=\lambda f(x,u)+\mu h(x)\quad &\text{in }\Omega,\\ u> 0\quad &\text{in }\Omega,\\ u=0\quad &\text{on }\partial\Omega,\end{cases}\tag{P\(_{\lambda,\mu}\)} \] where \(\lambda\) and \(\mu\) are positive parameters. Under suitable assumptions on \(K\), \(g\), \(f\) and \(h\) the authors using the maximum principle for elliptic equations establish several existence and nonexistence results for \((\text{P}_{\lambda,\mu})\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
58J55 Bifurcation theory for PDEs on manifolds
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aris, R., The mathematical theory of diffusion and reaction in permeable catalysts, (1975), Clarendon Press Oxford · Zbl 0315.76051
[2] Bénilan, P.; Brezis, H.; Crandall, M., A semilinear equation in \(L\^{}\{1\}( R\^{}\{N\})\), Ann. scuola norm. sup. cl. sci. Pisa, 4, 523-555, (1975) · Zbl 0314.35077
[3] Callegari, A.; Nashman, A., Some singular nonlinear equations arising in boundary layer theory, J. math. anal. appl., 64, 96-105, (1978) · Zbl 0386.34026
[4] Callegari, A.; Nashman, 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] Chen, H., On a singular nonlinear elliptic equation, Nonlinear anal. T.M.A., 29, 337-345, (1997) · Zbl 0882.35050
[6] Choi, Y.S.; Lazer, A.C.; McKenna, P.J., Some remarks on a singular elliptic boundary value problem, Nonlinear anal. T.M.A., 3, 305-314, (1998) · Zbl 0940.35089
[7] Coclite, M.M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. partial differential equations, 14, 1315-1327, (1989) · Zbl 0692.35047
[8] Cohen, D.S.; Keller, H.B., Some positive problems suggested by nonlinear heat generators, J. math. mech., 16, 1361-1376, (1967) · Zbl 0152.10401
[9] 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
[10] Diaz, J.I.; Morel, J.M.; Oswald, L., An elliptic equation with singular nonlinearity, Comm. partial differential equations, 12, 1333-1344, (1987) · Zbl 0634.35031
[11] Gomes, S.M., On a singular nonlinear elliptic problem, SIAM J. math. anal., 17, 1359-1369, (1986) · Zbl 0614.35037
[12] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer Berlin · Zbl 0691.35001
[13] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 3, 720-730, (1991) · Zbl 0727.35057
[14] Luning, C.D.; Perry, W.L., An interactive method for solution of a boundary value problem in non-Newtonian fluid flow, J. non-Newtonian fluid mech., 15, 145-154, (1984) · Zbl 0564.76013
[15] T. Ouyang, J. Shi, M. Yao, Exact multiplicity of positive solutions for a singular equation in unit ball, preprint, 2002.
[16] Perry, W.L., A monotone iterative technique for solution of pth order (p<0) reaction – diffusion problems in permeable catalysis, J. comput. chem., 5, 353-357, (1984)
[17] del Pino, M., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. roy. soc. Edinburgh sect. A, 122, 341-352, (1992) · Zbl 0791.35046
[18] Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. roy. soc. Edinburgh sect. A, 128, 1389-1401, (1998) · Zbl 0919.35044
[19] Wong, J., On the generalized emden – fowler equation, SIAM rev., 17, 339-360, (1975) · Zbl 0295.34026
[20] Zhang, Z., On a Dirichlet problem with a singular nonlinearity, J. math. anal. appl., 194, 103-113, (1995) · Zbl 0834.35054
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.