Multiple positive solutions for a second order Sturm-Liouville boundary value problem with a \(p\)-Laplacian via variational methods (Q1011146)

The authors study the Sturm-Liouville boundary value problem for a second order ordinary differential equation of \(p\)-Laplacian type. The aim of this paper is to provide the framework in which under the new types of growth assumptions on the nonlinear term the existence of at least two positive solutions can be obtained. The authors construct a suitable action functional, introduce the proper function spaces, i.e. weighted Sobolev spaces, whose properties, required in their work, they investigate. Such results, interesting in themselves, provide the background for further investigation of variational problems. The main results of this research concerns the existence of at least two positive classical solutions. One solution is obtained by the mountain pass approach and the other one through classical minimization in the ball. A nontrivial example finishes the paper.