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Global structure of positive solutions for superlinear discrete boundary value problems. (English) Zbl 1233.39006

The article deals with the following Dirichlet problem \[ \Delta^2u(t - 1) + \lambda a(t)f(u(t)) = 0, \qquad u(0) = u(T + 1) = 0,\eqno(1) \] where \(a:\;\{0,1,\ldots,T + 1\} \to {\mathbb R}^+\), \(f \in C([0,\infty),[0,\infty))\), \(f(s) > 0\) for \(s > 0\). Let \(f_0 = \lim_{s \to 0} \, f(s)/s = \infty\) and \(f_\infty = \lim_{s \to \infty} \, f(s)/s \in [0,\infty]\), \(\Sigma\) be the closure of set of positive solutions of Problem (1). The main results are the following statements: If \(f_\infty = 0\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) and \(\text{Proj}|_{\mathbb R} \, \zeta = (0,\infty)\); moreover, Problem (1) has at least one positive solution for \(\lambda \in (0,\infty)\); If \(f_\infty \in (0,\infty)\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) and \(\text{Proj}|_{\mathbb R} \, \zeta \subset \bigg(0,{\frac{\lambda_{1}}{f_{\infty}}}\bigg)\) moreover, Problem (1) has at least one positive solution for \(\lambda \in \bigg(0, {\frac{\lambda_{1}}{f_{\infty}}}\bigg)\) If \(f_\infty = \infty\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) for which \(\text{Proj}|_{\mathbb R} \, \zeta\) is a bounded closed interval and \(\zeta\)approaches \((0,\infty)\) as \(\|u\| \to \infty\) moreover, there exist \(0 < \lambda_*, \lambda^* < \infty\) such that Problem (1) has at least two positive solution for \(\lambda \in (0,\lambda_*\) and no positive solutions for \(\lambda > \lambda^*\). The number \(\lambda_1\) is the first (smallest) eigenvalue for the linear problem \[ \Delta^2u(t - 1) + \lambda a(t)u(t) = 0, \qquad u(0) = u(T + 1) = 0. \]

MSC:

39A12 Discrete version of topics in analysis
39A22 Growth, boundedness, comparison of solutions to difference equations
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