Positive solutions of a focal problem for one‐dimensional p‐Laplacian equations

This paper mainly deals with the existence and multiplicity of positive solutions for the focal problem involving both the $p$ ‐Laplacian and the first order derivative: $\{\begin{array}{c}{\left({\left(u^{\prime }\right)}^{p-1}\right)}^{\prime }+f(t,u,u^{\prime })=0\text{,}t\in (0,1)\text{,}\\ u\left(0\right)=u^{\prime }\left(1\right)=0\text{.}\end{array}$ The main tool in the proofs is the fixed point index theory, based on a priori estimates achieved by using Jensen’s inequality and a new inequality. Finally the main results are applied to establish the existence of positive symmetric solutions to the Dirichlet problem: $\{\begin{array}{c}{\left({\left|u^{\prime }\right|}^{p-2}{u}^{\prime }\right)}^{\prime }+f(u,u^{\prime })=0\text{,}t\in (-1,0)\cup (0,1)\text{,}\\ u(-1)=u\left(1\right)=0\text{.}\end{array}$