An existence and uniqueness result for the singular Lane-Emden-Fowler equation. (English) Zbl 1180.35256
Summary: Using a fixed point theorem of general \(\alpha \)-concave operators, we present criteria which guarantee the existence and uniqueness of positive solutions for the singular Lane-Emden-Fowler equation.
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35J61 | Semilinear elliptic equations |
| 35B09 | Positive solutions to PDEs |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
existence and uniqueness; positive solution; fixed point theorem of general \(\alpha \)-concave operators; singular Lane-Emden-Fowler equationReferences:
| [1] | Crandall, M. G.; Rabinowitz, P. H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2, 2, 193-222 (1977) · Zbl 0362.35031 |
| [2] | Fulks, W.; Maybee, J. S., A singular nonlinear elliptic equation, Osaka J. Math., 12, 1-19 (1960) · Zbl 0097.30202 |
| [3] | Ghergu, M.; Rădulescu, V., Sublinear singular elliptic problems with two parameters, J. Differential Equations, 195, 520-536 (2003) · Zbl 1039.35042 |
| [4] | Nachman, A.; Callegari, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 28, 275-281 (1980) · Zbl 0453.76002 |
| [5] | Stuart, C. A., Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147, 53-63 (1976) · Zbl 0324.35037 |
| [6] | Agarwal, R. P.; O’Regan, D., Singular boundary value problems for superlinear second order ordinary and delay differential equations, J. Differential Equations, 130, 333-335 (1996) · Zbl 0863.34022 |
| [7] | Agarwal, R. P.; O’Regan, D.; Lakshmikantham, V., Quadratic forms and nonlinear non-resonant singular second order boundary value problems of limit circle type, Z. Anal. Anwendungen, 20, 727-737 (2001) · Zbl 0986.34014 |
| [8] | Agarwal, R. P.; O’Regan, D., Existence theory for single and multiple solutions to singular positone boundary value problems, J. Differential Equations, 175, 393-414 (2001) · Zbl 0999.34018 |
| [9] | Agarwal, R. P.; O’Regan, D., Existence theory for singular initial and boundary value problems: A fixed point approach, Appl. Anal., 81, 391-434 (2002) · Zbl 1079.34010 |
| [10] | Coclite, M. M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14, 1315-1327 (1989) · Zbl 0692.35047 |
| [11] | Cui, S., Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems, Nonlinear Anal., 41, 149-176 (2000) · Zbl 0955.35026 |
| [12] | Dalmasso, R., Solution d’equations elliptiques semi-lineaires singulieres, Ann. Mat. Pura Appl., 153, 191-201 (1988) · Zbl 0692.35044 |
| [13] | Ghergu, M.; Rădulescu, V., (Singular Elliptic Problems. Bifurcation and Asymptotic Analysis. Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, vol. 37 (2008), Oxford University Press) · Zbl 1159.35030 |
| [14] | Gomes, S. M., On a singular nonlinear elliptic problem, SIAM J. Math. Anal., 17, 6, 1359-1369 (1986) · Zbl 0614.35037 |
| [15] | Hernández, J.; Karátson, J.; Simon, P. L., Multiplicity for semilinear elliptic equations involving singular nonlinearity, Nonlinear Anal., 65, 265-283 (2006) · Zbl 1387.35232 |
| [16] | Hernández, J.; Mancebo, F.; Vega, J. M., Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137, 41-62 (2007) · Zbl 1180.35231 |
| [17] | Lair, A. V.; Shaker, A. W., Classical and weak solutions of singular semilinear elliptic problem, J. Math. Anal. Appl., 211, 371-385 (1997) · Zbl 0880.35043 |
| [18] | Lazer, A. C.; Mckenna, P. J., On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111, 721-730 (1991) · Zbl 0727.35057 |
| [19] | Lin, X.; Jiang, D.; Li, X., Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. Comput. Appl. Math., 196, 155-161 (2006) · Zbl 1107.34307 |
| [20] | Magli, H.; Zribi, M., Existence and estimates of solutions for singular nonlinear elliptic problems, J. Math. Anal. Appl., 263, 522-542 (2001) · Zbl 1030.35064 |
| [21] | Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A Math., 128, 1389-1401 (1998) · Zbl 0919.35044 |
| [22] | Usami, H., On a singular elliptic boundary value problem in a ball, Nonlinear Anal., 13, 1163-1170 (1989) · Zbl 0707.35058 |
| [23] | Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear Anal., 28, 12, 1977-1995 (1997) · Zbl 0874.35061 |
| [24] | Zhang, Z. J., The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation, J. Math. Anal. Appl., 312, 33-43 (2005) · Zbl 1165.35377 |
| [25] | Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101 |
| [26] | Zhai, C. B.; Yang, C.; Guo, C. M., Positive solutions of operator equation on ordered Banach spaces and applications, Comput. Math. Appl., 56, 3150-3156 (2008) · Zbl 1165.47308 |
| [27] | Brezis, H.; Nirenberg, L., Minima locaux relatifs a \(C^1\) et \(H^1\), C. R. Acad. Sci., Paris, 317, 465-472 (1993) · Zbl 0803.35029 |
| [28] | Ladyzhenskaya, O. A.; Ural’ceva, N. N., Linear and Quasilinear Elliptic Equations (1968), Academic Press: Academic Press New York, (English transl.) · Zbl 0164.13002 |
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