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September  2015, 14(5): 1803-1816. doi: 10.3934/cpaa.2015.14.1803

## Positive solution for quasilinear Schrödinger equations with a parameter

 1 Business School of Hunan University, Changsha, Hunan 410082, China

Received  September 2014 Revised  February 2015 Published  June 2015

In this paper, we study the following quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{(2-\alpha)/2}}=\mathrm{g}(x,u)， \end{eqnarray} where $1 \le \alpha \le 2$, $N \ge 3$, $V\in C(R^N, R)$ and $\mathrm{g}\in C(R^N\times R, R)$. By using a change of variables, we get new equations, whose respective associated functionals are well defined in $H^1(R^N)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.
Citation: GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803
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