p-Kirchhoff type problem with a general critical nonlinearity

Abstract

In this article, we consider the p-Kirchhoff type problem $$ \Big(1+\lambda\int_{\mathbb{R}^N}|\nabla u|^p +\lambda b\int_{\mathbb{R}^N}|u|^p\Big)(-\Delta_p u+b|u|^{p-2}u) =f(u), x\in\mathbb{R}^N, $$ where $\lambda>0$, the nonlinearity f can reach critical growth. Without the Ambrosetti-Robinowitz condition or the monotonicity condition on f, we prove the existence of positive solutions for the p-Kirchhoff type problem. In addition, we also study the asymptotic behavior of the solutions with respect to the parameter $\lambda\to0$.

Keywords

• p-Kirchhoff type problem
• critical growth
• variational methods