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Abstract |
We consider the quasilinear Schr\"odinger equation \[-\Delta u+V(x)u-\Delta(u^{2})u=g(x,u), \quad x\in \mathbb{R}^{N},\] where $g$ and $V$ are periodic in $x_1,\ldots,x_N$, $V>0$, $g$ is odd in $u$ and of subcritical growth in the sense that $|g(x,u)|\leq a(1+|u|^{p-1})$ for some $4<p<2\cdot 2^*$. We show that this equation has infinitely many geometrically distinct solutions in each of the following two cases: \begin{itemize} \item[(i)] $g(x,u)=o(u)$ as $u\to 0$, $G(x,u)/u^4\to\infty$ as $|u|\to\infty$, where $G$ is the primitive of $g$, and $u\mapsto g(x,u)/u^3$ is positive for $u\ne 0$, nonincreasing on $(-\infty, 0)$ and nondecreasing on $(0, \infty)$. \item[(ii)] $g(x,u)=q(x)u^3$, where $q>0$.\end{itemize} The argument uses the Nehari manifold technique. A special feature here is that the Nehari manifold is not likely to be of class $C^1$. This is joint work with Xiangdong Fang. |
