# American Institute of Mathematical Sciences

November  2016, 15(6): 2457-2473. doi: 10.3934/cpaa.2016044

## Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential

 1 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received  March 2016 Revised  June 2016 Published  September 2016

In this paper, we study the existence, multiplicity and concentration of positive solutions for the following indefinite semilinear elliptic equations involving concave-convex nonlinearities: \begin{eqnarray} \left\{\begin{array}{l} -\Delta u+V_{\lambda }\left( x\right) u=f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\geq 0, & \text{in }\mathbb{R}^{N}, \end{array}\right. \end{eqnarray} where$1 < q < 2 < p < 2^{\ast} (2^{\ast } = \frac{2N}{N-2}$ for $N \geq 3)$ the potential $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $V^{\pm }=\max \left\{ \pm V,0\right\}$ and the parameter $\lambda >0.$ We assume that the functions $f,g$ and $V$ satisfy suitable conditions with the potential $V$ and the weight function $g$ without the assumptions of infinite limits.
Citation: Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
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