Kirchhoff-Type boundary-value problems on the real line

Abstract

This paper deals with the existence and energy estimates of positive solutions for a class of Kirchhoff-type boundary-value problems on the real line, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, applying a consequence of the local minimum theorem for differentiable functionals due to Bonanno the existence of a positive solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti–Rabinowitz. In what follows, employing two consequences of the local minimum theorem for differentiable functionals due to Bonanno by combining two algebraic conditions on the nonlinear term which guarantees the existence of two positive solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third positive solution for our problem. Moreover, concrete examples of applications are provided.