p-Kirchhoff type problem with a general critical nonlinearity - Huixing Zhang; Baiquan Lin (Electronic Journal of Differential Equations, 2018)
In this article, we consider the p-Kirchhoff type problem $$ \Big(1+λ\int_{\mathbb{R}^N}|\nabla u|^p +λ b\int_{\mathbb{R}^N}|u|^p\Big)(-Δ_p u+b|u|^{p-2}u) =f(u), x\in\mathbb{R}^N, $$ where $λ>0$, 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 $λ→0$.