Positive solution for quasilinear Schrödinger equations with a parameter

GUANGBING LI

doi:10.3934/cpaa.2015.14.1803

Article Contents

2015, Volume 14, Issue 5: 1803-1816. Doi: 10.3934/cpaa.2015.14.1803

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Positive solution for quasilinear Schrödinger equations with a parameter

GUANGBING LI^1,

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

Received: September 2014

Revised: February 2015

Published: September 2015

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Abstract

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.

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Mathematics Subject Classification: 35J20, 35J60, 35Q55.

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