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Positive solutions of singular sublinear Dirichlet boundary value problems. (English) Zbl 0823.34031

This paper mainly concerns singular sublinear boundary value problems in the interval \(I = (0,1)\), of type (1) \(- \ddot u = f(t,u)\), \(t \in I\), \(u(0) = 0 = u(1)\), where \(f\) is a nonnegative continuous function in \(I \times (0, \infty)\) with \(f(t,1)\) not identically zero, and \(f(t,u)\) satisfies sublinearity conditions with respect to \(u\). The main theorems are necessary and sufficient conditions for (1) to have a positive solution \(u \in C(\overline I)\) or \(u \in C^ 1 (\overline I)\). The proofs employ a barrier method. Related results were obtained by L. E. Bobisud, D. O’Regan and W. D. Royalty [Appl. Anal. 23, 233-243 (1986; Zbl 0595.34013)], D. O’Regan [J. Math. Anal. Appl. 142, 40-52 (1989; Zbl 0689.34015)], and the author [Nonlinear Anal., Theory Methods Appl. 21, 153-159 (1993; Zbl 0790.34027); J. Math. Anal. Appl. 185, 215-222 (1994)]. Also a uniqueness theorem is proved for (1), and applications to radially symmetric elliptic Dirichlet problems are included.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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