## Positive solutions for nonlinear eigenvalue problems.(English)Zbl 0876.34023

The authors are concerned with determining values of $$\lambda$$ (eigenvalues), for which there exist positive solutions of the boundary value problem $(1_\lambda)\quad u''+\lambda a(t)f(u)=0,\;0<t<1, \qquad (2) \quad u(0)= u(1)=0,$ where $$f:[0,\infty)\to [0,\infty)$$ is continuous; $$a:[0,1]\to [0,\infty)$$ is continuous and does not vanish identically on any subinterval, and $f_0= \lim_{x\to 0} (f(x)/x), \qquad f_\infty= \lim_{x\to\infty} (f(x)/x)$ exist. To research the problem $$(1_\lambda)$$, (2) the Krasnosel’skij methods of solutions of nonlinear operator equations in a space with a cone are applicable.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations 34B24 Sturm-Liouville theory 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
Full Text:

### References:

 [1] Bandle, C.; Coffman, C. V.; Marcus, M., Nonlinear elliptic problems in annular domains, J. Differential Equations, 69, 322-345 (1987) · Zbl 0618.35043 [2] Bandle, C.; Kwong, M. K., Semilinear elliptic problems in annular domains, J. Appl. Math. Phys., 40, 245-257 (1989) · Zbl 0687.35036 [3] Chyan, C. J.; Henderson, J., Positive solutions for singular higher order nonlinear equations, Differential Equations Dynam. Systems, 2, 153-160 (1994) · Zbl 0873.34013 [4] Eloe, P. W.; Henderson, J., Positive solutions for higher order differential equations, Elec. J. Differential Equations, 3, 1-8 (1995) [5] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018 [6] Erbe, L. H.; Wang, H., Existence and nonexistence of positive solutions in annular domains, WSSIAA, 3, 207-217 (1994) · Zbl 0900.35144 [7] de Figueiredo, D. G.; Lions, P. L.; Nussbaum, R. D., A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pura Appl., 61, 41-63 (1982) · Zbl 0452.35030 [8] Fink, A. M., The radial Laplacian Gel’fand problem, Delay and Differential Equations (1992), World Scientific: World Scientific River Edge, p. 93-98 · Zbl 0820.34049 [9] Fink, A. M.; Gatica, J. A.; Hernandez, G. E., Eigenvalues of generalized Gel’fand models, Nonlinear Anal., 20, 1453-1468 (1993) · Zbl 0790.34021 [10] Garaizer, X., Existence of positive radial solutions for semilinear elliptic problems in the annulus, J. Differential Equations, 70, 69-72 (1987) [11] Keller, H. B., Some positive problems suggested by nonlinear heat generation, (Keller, J. B.; Antman, S., Bifurcation Theory and Nonlinear Eigenvalue Problems (1969), Benjamin: Benjamin Elmsford), 217-255 [12] Kuiper, H. J., On positive solutions of nonlinear elliptic eigenvalue problems, Rend. Circ. Mat. Palermo (2), 20, 113-138 (1979) · Zbl 0263.35028 [13] Sanchez, L., Positive solutions for a class of semilinear two-point boundary value problems, Bull. Austral. Math. Soc., 45, 439-451 (1992) · Zbl 0745.34017 [14] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen · Zbl 0121.10604 [15] Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations, 109, 1-7 (1994) · Zbl 0798.34030
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.