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Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line. (English) Zbl 1346.34020

Summary: We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem \[ \frac{1}{A}(Au')'+a_1(t)u^{\sigma_{1}}+a_2(t)u^{\sigma_{2}}=0, \quad t\in (0, \infty), \] subject to the boundary conditions \(\lim_{t\to 0^{+}} u(t)=0\), \(\lim_{t\to\infty}u(t)/\rho(t)=0\), where \(\sigma_{1},\sigma_{2}<1\) and \(A\) is a continuous function on \([0,\infty)\) which is positive and differentiable on \((0,\infty)\) such that \(\int_{0}^{1}1/A(t)\,dt<\infty\) and \(\int_{0}^{\infty}1/A(t)\,dt=\infty\). Here, \(\rho(t)=\int_{0}^{t}1/{A(s)}\,ds\) for \(t>0\) and \(a_{1},a_{2}\) are nonnegative continuous functions on \((0,\infty)\) that may be singular at \(t=0\) and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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