Positive solutions for equations and systems with \(p\)-Laplace-like operators. (English) Zbl 1185.34028

García-Huidobro, Marta; Manásevich, Raul; Ward, James R.

Adv. Differ. Equ. 14, No. 5-6, 401-432 (2009).

The authors consider the existence of positive solutions of \[ \big( \phi(u') \big)' + f(t, u) = 0, \quad t \in (0, 1), \]
\[ \theta(u(0)) = \beta \theta (u'(0)), \]
\[ \theta(u(1)) = -\delta \theta (u'(1)), \] where \(\beta, \delta \geq 0\), as well as related boundary value problems. Here, \(\phi\) and \(\theta\) are odd increasing homeomorphisms and \(f:[0, 1] \times [0, \infty) \to [0, \infty)\) is continuous. They prove three theorems on existence using a maximum principle and the Leray-Schauder degree. In each theorem, they assume that the nonlinear term \(f\) is either super-linear or sub-linear. Additionally, they need to assume that limits involving ratios of the form \(\frac{\theta(\tau s)}{\theta(s)}\) are positive and tend to either \(0\) or \(\infty\).

Reviewer: Eric R. Kaufmann (Little Rock)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

Leray-Schauder degree; \(p\)-Laplacian; positive solution