Positive solutions of one-dimensional $p$-Laplacian equations and applications to population models of one species

Abstract

We prove new results on the existence ofpositive solutions of one-dimensional $p$-Laplacian equationsunder sublinear conditions involving the first eigenvalues of the corresponding homogeneous Dirichlet boundary value problems. To the best of our knowledge, this is the first paper to use fixed point index theory of compact maps to give criteriainvolving the first eigenvalue for one-dimensional $p$-Laplacian equations with $p\ne 2$.Our results generalize some previous results where either $p$ is required to be greater than $2$ or the nonlinearities satisfy stronger conditions.We shall apply our results to tackle a logistic population modelarising in mathematical biology.