Electron. J. Differential Equations, Vol. 2017 (2017), No. 292, pp. 1-11.

Positive solutions for p-Laplacian equations of Kirchhoff type problem with a parameter

Qi Zhang, Jianping Huang

Abstract:
In this article, we consider the existence and non-existence of positive solutions for the Kirchhoff type equation
$$\displaylines{ -\Big(a+\lambda M \Big(\int_{\Omega}|\nabla u|^{p}dx\Big)\Big)\Delta_pu= f(u), \quad \text{in } \Omega,\cr u=0, \quad \text{on } \partial\Omega, }$$
where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with a smooth boundary $\partial\Omega$, a is a positive constant, $N\geq 3$, $\lambda\geq0$, $2\leq p<N$, M and f are positive continuous functions. Under some weak assumptions on f, we show that the above problem has at least one positive solution when $\lambda$ is small and has no nonzero solution when $\lambda$ is large. Our argument is based on iterative technique and variational methods.

Submitted September 11, 2017. Published November 27, 2017.
Math Subject Classifications: 35B30, 35B09, 35J50, 35J62.
Key Words: Positive solution; p-Laplacian equation; iterative technique.

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Qi Zhang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: zq8910@csu.edu.cn
Jianping Huang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: huangjianping@csu.edu.cn

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