Multiplicity and nonexistence of positive solutions to impulsive Sturm–Liouville boundary value problems

In this paper, we study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear impulsive Sturm–Liouville boundary value problem with a parameter. By using a variational method, we prove that the problem has at least two positive solutions for the parameter $\lambda \in (0,\mathrm{\Lambda })$ $\lambda \in (0,\Lambda )$ , one positive solution for $\lambda =\mathrm{\Lambda }$ $\lambda =\Lambda $ , and no positive solution for $\lambda >\mathrm{\Lambda }$ $\lambda >\Lambda $ , where $\mathrm{\Lambda }>0$ $\Lambda >0$ is a constant.