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Analysis of positive solutions for a class of semipositone \(p\)-Laplacian problems with nonlinear boundary conditions. (English) Zbl 1418.34057

The authors consider the existence and nonexistence of solutions to the singular \(p\)-Laplacian boundary value problem \[ -((\varphi_p(u'))' = \lambda h(t) \frac{f(u)}{u^\alpha}, \quad 0 < t < 1, p > 1, \] \[ u'(1) + c (u(1))u(1) = 0, \] \[ u(0) = 0. \] Here \(\varphi(u) = |u|^{p-2}, \lambda > 0, 0 \leq \alpha < 1, c:[0, +\infty) \to [0, +\infty)\) is continuous, and \(h:(0, 1) \to (0, +\infty)\) is continuous and integrable. The paper is divided into four sections. In Section 2, the authors show that under suitable conditions if \(\psi\) is a subsolution and \(\varphi\) is a supersolution of the boundary value problem such that \(\psi \leq \varphi\) on \([0, 1]\) then the boundary value problem has at least one solution \(u \in C^1[0, 1]\) such that \(\psi \leq u \leq \varphi\) on \([0, 1]\). In Section 3, the authors first establish a nonexistence theorem for the boundary value problem when \(\lambda \approx 0\). They then construct subsolutions and supersolutions of the boundary value problem when \(\lambda \gg 1\) to establish that there exists a positive solution \(u\) such that \(\|u\| \to \infty\) as \(\lambda \to \infty\). Finally, in Section 4, the authors show that under suitable conditions, if \(\alpha = 0\) and \(\lambda \gg 1\) then the boundary value problem has a unique solution.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
Full Text: DOI

References:

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