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

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 $$\lambda\to \infty$$.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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