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

In this article, we consider the existence of multiple solutions to the elliptic problem
 -\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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