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

In this paper, we study a singular Kirchhoff-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 bounded on the associated Nehari manifold and the associated energy functional is not differentiable because of the singular term, we cannot apply the variational method according to a standard way. By analyzing the structure of the Nehari manifold and developing some approximation techniques, the above obstacles are overcome in subtle ways. As a result, two positive solutions of that problem are obtained with negative and positive energy, respectively.