Multiplicity of positive solutions for a nonlinear fourth order equation (Q2759189)

The author is concerned with the existence and multiplicity of positive solutions to the nonlinear fourth-order boundary value problem NEWLINE\[NEWLINEu^{( 4)}=\lambda f(u)\text{ in }(0,1), \qquad u(0)=a, u'(0)=a',\;u(1)=b,u'(1)=-b',\tag \(1_\lambda\) NEWLINE\]NEWLINE 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 \textit{H. Dang}, \textit{K. Schmitt} and \textit{R. Shivaji} [Electron J. Differ. Equ. 1996, No. 1 (1996; Zbl 0848.35050)].