Positive solutions for p ‐Kirchhoff type problems on R N

In this paper, we establish the existence and non‐existence of positive solutions for p ‐Kirchhoff type problems with a parameter onR Nwithout assuming the usual compactness conditions. We show that the p ‐Kirchhoff type problems have at least one positive solution when the parameter is small, while the p ‐Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut‐off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.