Multiplicity and concentration of positive solutions for semilinearelliptic equations with steep potential

Yi-hsin  Cheng; Tsung-Fang Wu

doi:10.3934/cpaa.2016044

Article Contents

2016, Volume 15, Issue 6: 2457-2473. Doi: 10.3934/cpaa.2016044

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

Yi-hsin Cheng^1, and
Tsung-Fang Wu^1,

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

Received: February 29, 2016

Revised: May 31, 2016

Published: October 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J65.

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