Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity

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• 作者：Hongxia Shi ^{shihongxia5617@163.com" class="auth_mail" title="E-mail the corresponding author} ; Haibo Chen ; ^{math_chb@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quasilinear Schrö ; dinger equations ; Vanishing potential ; Variational methods
• 刊名：Applied Mathematics Letters
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：61
• 期：Complete
• 页码：137-142
• 全文大小：564 K

文摘

This paper is concerned with the following quasilinear Schrödinger equations: where N≥3$N\ge 3$ and V$V$, K$K$ are nonnegative continuous functions. Firstly by using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is still not well defined in D^1,2(R^N)${\mathcal{D}}^{1,2}\left({\mathbb{R}}^N\right)$ because of the potential vanishing at infinity. However, by using a Hardy-type inequality, we can work in the weighted Sobolev space in which the functional is well defined. Using this fact together with the variational methods, we obtain a positive solution.