Positive solutions of a second-order boundary value problem via integro-differential equation arguments

Abstract

The objective of this article is to establish the existence and multiplicity of positive solutions for the second-order boundary value problem depending on the first-order derivative u′:

Our way to overcome the difficulty resulting from the first-order derivative is to transform it into an initial problem of a first-order integro-ordinary differential equation. Then, based on a priori estimates of positive solutions of an auxiliary problem, we use index theory to establish the existence and multiplicity of positive solutions for the above problem. The main results are used to establish the existence of symmetric positive solutions for the Dirichlet problem.

Keywords:

• positive solution
• integro-ordinary differential equation
• Dirichlet problem
• symmetric positive solution

AMS Subject Classifications::

• 34B15
• 45J05
• 34L15
• 47H11
• 45M20

Acknowledgements

This work was supported by the NNSF of China (Grant 10871116) and the Science and Technology Project of the Education Department of Shandong Province of China (Grant No. J0P605).