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

DOI: http://dx.doi.org/10.12775/TMNA.2015.053

#### Abstract

We prove new results on the existence of positive solutions of one-dimensional $p$-Laplacian equations under 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 criteria involving 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 model arising in mathematical biology.

#### Keywords

One-dimensional $p$-Laplacian equations,positive solutions, sublinear condition,fixed point index, logistic population model

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