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Nontrivial solutions of singular superlinear Sturm--Liouville problems. (English) Zbl 1100.34019
The authors study the singular superlinear problem $$-(p(x)y')'-q(x)y=h(x)f(y),\quad 0<x<1,$$ $$\alpha_1 y(0)+\beta_1 y'(0)=0,\quad \alpha_2 y(1)+\beta_2 y'(1)=0.$$ The function $h$ is allowed to be singular at both $x=0$ and $x=1$. In addition, $f$ is not assumed to be nonnegative. The assumption of nonnegativity of $f$ has been very often required in the existing literature. Omitting this condition requires a different approach. Using topological degree theory, the authors establish conditions guaranteeing the existence of nontrivial solutions and positive solutions to the above boundary value problem. A nonsingular case is discussed as well.

MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE
Full Text:
References:
 [1] O’regan, D.: Theory of singular boundary value problems. (1994) [2] Habets, P.; Zanolin, F.: Upper and lower solutions for a generalized Emden -- Fowler equation. J. math. Anal. appl. 181, 684-700 (1994) · Zbl 0801.34029 [3] Erbe, L. H.; Mathsen, R. M.: Positive solutions for singular nonlinear boundary value problems. Nonlinear anal. 46, 979-986 (2001) · Zbl 1007.34020 [4] Wang, J.: On positive solutions of singular nonlinear two-point boundary value problems. J. differential equations 107, 163-174 (1994) · Zbl 0792.34023 [5] Zhang, Y.: Positive solutions of singular sublinear Emden -- Fowler boundary value problems. J. math. Anal. appl. 185, 215-222 (1994) · Zbl 0823.34030 [6] Ma, R.: Positive solutions of singular second order boundary value problem. Acta math. Sinica 41, 1225-1230 (1998) · Zbl 1027.34025 [7] Wei, Z.: Positive solutions of singular boundary value problems of negative exponent Emden -- Fowler equations. Acta math. Sinica 41, 655-662 (1998) · Zbl 1027.34024 [8] Zhao, Z.: Positive solutions of boundary value problems for nonlinear singular differential equations. Acta math. Sinica 43, 179-188 (2000) · Zbl 1018.34018 [9] Cheng, J.: Positive solutions of second order boundary value problems. Acta math. Sinica 44, 429-436 (2001) · Zbl 1018.34020 [10] Agarwal, R. P.; O’regan, D.: A note on existence of nonnegative solutions to singular semi-positone problems. Nonlinear anal. 36, 615-622 (1999) · Zbl 0921.34027 [11] Cheng, J.: Positive solutions of singular semi-positone problems. Acta math. Sinica 44, 673-678 (2001) · Zbl 1024.34017 [12] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045 [13] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [14] Guo, D.; Sun, J.; Liu, Z.: Functional methods in nonlinear ordinary differential equations. (1995) [15] Krasnoselskii, M. A.; Zabreiko, P. P.: Geometrical methods of nonlinear analysis. (1984) [16] Rubinstein, Z.: A course in ordinary and partial differential equations. (1969) · Zbl 0175.37801 [17] Guo, D.; Sun, J.: Nonlinear integral equations. (1987) [18] Krasnoselskii, M. A.: Topological methods in the theory of nonlinear integral equations. (1964) [19] Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations. (1967) · Zbl 0153.13602 [20] Li, Y.: Positive solutions of second-order boundary value problems with sign-changing nonlinear terms. J. math. Anal. appl. 282, 232-240 (2003) · Zbl 1030.34023