Electron. J. Diff. Equ., Vol. 2015 (2015), No. 78, pp. 1-11.

Multiple positive solutions for elliptic problem with concave and convex nonlinearities

Jiayin Liu, Lin Zhao, Peihao Zhao

Abstract:
In this article, we consider the existence of multiple solutions to the elliptic problem
$$\displaylines{ -\Delta u=\lambda u^q+u^s+\mu u^p\quad \text{in } \Omega,\cr u>0\quad \text{in } \Omega,\cr u=0\quad \text{on } \partial\Omega, }$$
where $\Omega\subseteq \mathbb{R}^N (N\geq3)$ is a bounded domain with smooth boundary $\partial\Omega$, $0<q<1<s<2^*-1\leq p$, $2^*:=\frac{2N}{N-2}$, $\lambda$ and $\mu$ are nonnegative parameters. By using variational methods, truncation and Moser iteration techniques, we show that if the parameters $\lambda$ and $\mu$ are small enough, then the problem has at least two positive solutions.

Submitted November 27, 2014. Published March 31, 2015.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Variational methods; supercritical exponent; mountain pass theorem, Moser iteration technique.

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Jiayin Liu
School of Mathematics and Statistics, Lanzhou University
Lanzhou, Gansu 730000, China
email: xecd@163.com
Lin Zhao
Department of Mathematics
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: zhaolinmath@gmail.com
Peihao Zhao
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: zhaoph@lzu.edu.cn

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