Kim, Chan-Gyun; Lee, Yong-Hoon Existence and multiplicity results for nonlinear boundary value problems. (English) Zbl 1142.34319 Comput. Math. Appl. 55, No. 12, 2870-2886 (2008). Summary: This paper studies the existence of positive solutions for a class of boundary value problems of elliptic degenerate equations. By using bifurcation and fixed point index theories in the frame of approximation arguments, the criteria of the existence, multiplicity and nonexistence of positive solutions are established. Cited in 5 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B09 Boundary eigenvalue problems for ordinary differential equations Keywords:existence; multiplicity; nonexistence; elliptic degenerate problems; positive solution; singular indefinite weight; global continuation theorem; approximation argument PDFBibTeX XMLCite \textit{C.-G. Kim} and \textit{Y.-H. Lee}, Comput. Math. Appl. 55, No. 12, 2870--2886 (2008; Zbl 1142.34319) Full Text: DOI References: [1] Asakawa, H., Nonresonant singular two-point boundary value problems, Nonlinear Anal., 47, 4849-4860 (2001) · Zbl 1042.34526 [2] Berestycki, B.; Dias, J. P.; Esteban, M. J.; Figuira, M., Eigenvalue problems for some nonlinear Wheeler-DeWitt operator, J. Math. Pures Appl., 72, 493-515 (1993) · Zbl 0839.35096 [3] Agarwal, R. P.; Lü, H.; O’Regan, D., Eigenvalues and the one-dimensional \(p\)-Laplacian, J. Math. Anal. Appl., 266, 383-400 (2002) · Zbl 1002.34019 [4] Choi, Y. S., A singular boundary value problem arising from near-ignition analysis of flame structure, Differential and Integral Equations, 4, 891-895 (1991) · Zbl 0795.34015 [5] Dalmasso, R., Positive solutions of singular boundary value problems, Nonlinear Anal., 27, 645-652 (1996) · Zbl 0860.34008 [6] Ha, K. S.; Lee, Y. H., Existence of multiple positive solutions of singular boundary value problems, Nonlinear Anal., 28, 1429-1438 (1997) · Zbl 0874.34016 [7] Lee, Y. H., Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains, Differential and Integral Equations, 13, 631-648 (2000) · Zbl 0970.35036 [8] Wong, F. H., Existence of positive solutions of singular boundary value problems, Nonlinear Anal., 21, 397-406 (1993) · Zbl 0790.34026 [9] Xu, X.; Ma, J., A note on singular nonlinear boundary value problems, J. Math. Anal. Appl., 293, 108-124 (2004) · Zbl 1057.34007 [10] Im, B. H.; Lee, E. K.; Lee, Y. H., A global bifurcation phenomenon for second order singular boundary value problems, J. Math. Anal. Appl., 308, 61-78 (2005) · Zbl 1091.34013 [11] Lan, K.; Webb, J. R.L., Positive solution of semilinear differential equations with singularities, J. Differential Equations, 148, 407-421 (1998) · Zbl 0909.34013 [12] Lee, E. K.; Lee, Y. H., Multiplicity results of positive solutions for singular boundary value problems with time dependent nonlinearity, Dynam. Systems Appl., 16, 233-250 (2007) · Zbl 1153.34012 [13] Liu, X.; Feng, W.; Liu, H., Non-resonance and eigenvalues of nonlinear singular boundary value problems, Nonlinear Anal., 50, 995-1012 (2002) · Zbl 1011.34016 [14] Mao, A.; Luan, S.; Ding, Y., On the existence of positive solutions for a class of singular boundary value problems, J. Math. Anal. Appl., 298, 57-72 (2004) · Zbl 1073.34019 [15] Berestycki, H.; Esteban, M. J., Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134, 1-25 (1997) · Zbl 0870.34032 [16] Cheng, J.; Zhang, Z., On the existence of positive solutions for a class of singular boundary value problems, Nonlinear Anal., 44, 645-655 (2001) · Zbl 0998.34019 [17] Lee, Y. H., An existence result of positive solutions for singular superlinear boundary value problems and its application, J. Korean Math. Soc., 34, 247-255 (1997) · Zbl 0879.34033 [18] Stanćzy, R., Positive solutions for superlinear elliptic equations, J. Math. Anal. Appl., 283, 159-166 (2003) · Zbl 1093.35027 [19] C.G. Kim, Y.H. Lee, Existence of multiple positive solutions for \(p\)-Laplacian problems with a general indefinite weight, Commun. Contemp. Math. (in press); C.G. Kim, Y.H. Lee, Existence of multiple positive solutions for \(p\)-Laplacian problems with a general indefinite weight, Commun. Contemp. Math. (in press) · Zbl 1158.34316 [20] do Ó, J.; Lorca, S.; Sánchez, J.; Ubilla, P., Non-homogeneous elliptic equations in exterior domains, Proc. Roy. Soc. Edinburgh, 136 A, 139-147 (2006) · Zbl 1281.35041 [21] C.G. Kim, Y.H. Lee, Existence of multiple positive solutions for two-point boundary value problem, preprint; C.G. Kim, Y.H. Lee, Existence of multiple positive solutions for two-point boundary value problem, preprint [22] Zeidler, E., Nonlinear Functional Analysis and its Applications I (1985), Springer-Verlag: Springer-Verlag New York [23] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 [24] Brezis, H.; Marcus, M., Hardy’s inequalities revisited, Ann. Sc. Norm. Super. Pisa, 25, 217-237 (1997) · Zbl 1011.46027 [25] De Coster, C.; Grossinho, M. R.; Habets, P., On Pairs of Solutions for a Singular Boundary Value, Appl. Anal., 59, 241-256 (1995) · Zbl 0847.34022 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.