Electron. J. Differential Equations, Vol. 2018 (2018), No. 89, pp. 1-10.

p-Kirchhoff type problem with a general critical nonlinearity

Huixing Zhang, Baiquan Lin

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 $\lambda\to0$.

Submitted September 5, 2017. Published April 11, 2018.
Math Subject Classifications: 35B25, 35B33, 35J61.
Key Words: p-Kirchhoff type problem; critical growth; variational methods.

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Huixing Zhang
School of Mathematics and
School of Safety Engineering
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: huixingzhangcumt@163.com
  Baiquan Lin
School of Safety Engineering
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: lbq21405@126.com

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