Multiple positive solutions for a singular Kirchhoff-type problem with convex nonlinearity on unbounded domain

Abstract

In this paper, we study a singularKirchhoff-type problem with a nonlinearity$h(x)|u|^{q-2} u$$(2 < q < 4)$on an unbounded domain.Since the (PS) sequence may not be boundedon the associated Nehari manifold and theassociated energy functional is notdifferentiable because of the singular term,we cannot apply the variationalmethod according to a standard way.By analyzing the structure of theNehari manifold and developingsome approximation techniques,the above obstacles are overcome insubtle ways. As a result,two positive solutions of that problemare obtained with negative and positive energy,respectively.