Multiplicity of positive solutions for a fourth-order quasilinear singular differential equation

Zhichang Guo, Jilin University, Changchun, P. R. China
Jingxue Yin, South China Normal University, Guangzhou, P. R. China
Yuanyuan Ke, Renmin University of China, Beijing, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 27. (2010), pp. 1-15.

Abstract: This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation
$$
(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,
$$
where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$.

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