## 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

Zbl 0848.35050
Full Text: