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Upper and lower solutions method and a superlinear singular boundary value problem. (English) Zbl 1058.34021
The paper is devoted to the solvability of a singular Dirichlet boundary value problem of the form $$ u''= g(t,u,u'),\quad u(0)=u(1)=0. \tag1$$ The nonlineearity $g: (0,1)\times (0,\infty)\times \bbfR \to \bbfR$ is continuous, can change its sign and can have time singularities at $t=0$ and $t=1$ and a space singularity at $x=0$. Moreover, $g$ may be superlinear in its second variable $x$ for $x\to \infty$ and sublinear in its third variable $y$ for $\vert y\vert \to \infty$. The existence of a solution $u\in C[0,1]\cup C^2(0,1)$ such that $u(t)>0$ for $t\in (0,1)$ and $t(1-t)u'\in C[0,1]$ is proved. The proof is based on the Leray-Schauder fixed-point theorem and on the lower and upper solutions method. The author was motivated by {\it P. Habets} and {\it F. Zanolin} [J. Math. Anal. Appl. 181 No.3, 684-700 (1994; Zbl 0801.34029)] and by {\it R.P. Agarwal} and {\it D. O’Regan} [J. Differ. Equations 143 No.1, 60-95 (1998; Zbl 0902.34015)].

34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
Full Text: DOI
[1] Agarwal, R. P.; O’regan, D.: Singular boundary value problems for superlinear second ordinary and delay differential equations. J. differential equations 130, 335-355 (1996)
[2] Agarwal, R. P.; O’regan, D.: Nonlinear superlinear singular and nonsingular second order boundary value problems. J. differential equations 143, 60-95 (1998) · Zbl 0902.34015
[3] Bobisud, L.; O’regan, D.; Royalty, W. D.: Singular boundary value problem. Applicable analysis 23, 233-243 (1986) · Zbl 0584.34012
[4] Bobisud, L.; O’regan, D.; Royalty, W. D.: Solvability of some nonlinear boundary value problems. Non-linear analysis 12, 855-869 (1988) · Zbl 0653.34015
[5] Callegari, A.; Nachman, A.: Some singular nonlinear differential equations arising in boundary layer theory. J. math. Anal. appl. 64, 96-105 (1978) · Zbl 0386.34026
[6] Gatica, J.; Hernandez, G.; Waltman, P.: Radially symmetric solutions of a class of singular elliptic equations. Proc. Edinburgh math. Soc. 33, 168-180 (1990) · Zbl 0689.35029
[7] Gatica, J.; Oliker, V.; Waltman, P.: Singular nonlinear boundary value problems for second order differential equations. J. differential equations 79, 62-78 (1989) · Zbl 0685.34017
[8] Gomes, S. M.; Sprekels, J.: Krasonselskii’s theorem on operators compressing a cone: application to some singular boundary value problems. J. math. Anal. appl. 153, 443-459 (1990) · Zbl 0766.47033
[9] Janus, J.; Myjak, J.: A generalized Emden-Fowler equation with a negative exponent. Nonlinear analysis 23, 953-970 (1994) · Zbl 0819.34016
[10] 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
[11] Jiang, D.; Liu, H.: On the existence of nonnegative radial solutions for the one-dimensional p-Laplacian elliptic systems. Ann. polon. Math. 71, 19-29 (1999) · Zbl 0928.34021
[12] O’regan, D.: Positive solutions to singular and nonsingular second-order boundary value problems. J. math. Anal. appl. 142, 40-52 (1989)
[13] O’regan, D.: Singular dirchlet boundary value problems--I, superlinear and nonresonant case. Nonlinear analysis 29, 221-245 (1997)
[14] Taliaferro, S.: A nonlinear singular boundary value problem. Nonlinear analysis 3, 897-904 (1979) · Zbl 0421.34021
[15] Wang, J.: A two-point boundary value problem with singularity. Northeast. math. J. 3, 281-291 (1987) · Zbl 0669.34023
[16] Wang, J.: On positive solutions of singular nonlinear two-point boundary problems. J. differential equations 107, 163-174 (1994) · Zbl 0792.34023
[17] Wang, J.; Gao, W.: A singular boundary value problem for the one-dimensional p-Laplacian. J. math. Anal. appl. 201, 851-866 (1996) · Zbl 0860.34011
[18] Wang, J.; Jiang, J.: The existence of positive solutions to a singular nonlinear boundary value problem. J. math. Anal. appl. 176, 322-329 (1993) · Zbl 0781.34018
[19] Jiang, D. Q.: Upper and lower solutions for a superlinear singular boundary value problem. Computers math. Applic. 41, No. 5/6, 563-569 (2001) · Zbl 0991.34018