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Multiplicity of positive solutions for a nonlinear fourth order equation. (English) Zbl 0989.34014

The author is concerned with the existence and multiplicity of positive solutions to the nonlinear fourth-order boundary value problem \[ u^{( 4)}=\lambda f(u)\text{ in }(0,1), \qquad u(0)=a, u'(0)=a',\;u(1)=b,u'(1)=-b',\tag \(1_\lambda\) \] where \(\lambda\) is a parameter, \(a,b,a',b'\) are nonnegative constants. The main results are that if \(f\in C([0,\infty), [0,\infty))\) is nondecreasing and \(\lim_{u\to 0}{f(u)\over u}=\lim_{u\to\infty} {f(u)\over u}= \infty\), then there exists \(\lambda^*\) such that \((1_\lambda)\) has at least two positive solutions for \(0<\lambda <\lambda^*\), at least one positive solution for \(\lambda=\lambda^*\) and no solutions for \(\lambda> \lambda^*\). The methods employed are upper and lower solutions and degree arguments. For related results on the second-order \(p\)-Laplacian, see H. Dang, K. Schmitt and R. Shivaji [Electron J. Differ. Equ. 1996, No. 1 (1996; Zbl 0848.35050)].
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0848.35050
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