Positive solutions of a focal problem for one-dimensional pp-Laplacian equations

This paper mainly deals with the existence and multiplicity of positive solutions for the focal problem involving both the pp-Laplacian and the first order derivative: {((u′)p−1)′+f(t,u,u′)=0,t∈(0,1),u(0)=u′(1)=0. 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: {(|u′|p−2u′)′+f(u,u′)=0,t∈(−1,0)∪(0,1),u(−1)=u(1)=0.