Advances in Differential Equations

$G$-invariant positive solutions for a quasilinear Schrödinger equation

Shinji Adachi and Tatsuya Watanabe

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We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.

Article information

Adv. Differential Equations, Volume 16, Number 3/4 (2011), 289-324.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 58E4


Adachi, Shinji; Watanabe, Tatsuya. $G$-invariant positive solutions for a quasilinear Schrödinger equation. Adv. Differential Equations 16 (2011), no. 3/4, 289--324.

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