Positive solutions of quasilinear boundary value problems. (English) Zbl 0893.34017

Hai, D. D.; Schmitt, K.; Shivaji, R.

J. Math. Anal. Appl. 217, No. 2, 672-686 (1998).

Two-point boundary value problems are considered \((p(t) \varphi(u'))' +\lambda p(t) \cdot f(t,u)=0\), \(a<t <b\) \(u(a)=u(b) =0\) (1). On assumption that 1) \(f:[a,b] \times \mathbb{R}\to \mathbb{R}\) is continuous, 2) \(p:[a,b] \to(0, \infty)\) is continuous, 3) \(f\) is either \(\varphi\)-superlinear or \(\varphi\)-sublinear at \(\infty\) it is proved that a) there exists \(\lambda^* >0\) such that problem (1) has a positive solution \(\mu_\lambda\) for \(0< \lambda< \lambda^*\) with \(\| u_\lambda \|_\infty \to\infty\) as \(\lambda\to 0\); b) there exists \(\overline \lambda>0\) such that problem (1) has a positive solution \(u_\lambda\) for \(\lambda> \overline\lambda\) with \(\| u_\lambda \|_\infty\to \infty\) as \(\lambda\to \infty\).

Reviewer: A.I.Kolosov (Khar’kov)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

\(\varphi\)-superlinear; \(\varphi\)-sublinear functions; positive solutions; quasilinear boundary value problems

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