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| Format: | Preprint |
| Veröffentlicht: |
2019
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| Online-Zugang: | https://arxiv.org/abs/1904.03520 |
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| _version_ | 1866916290811133952 |
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| author | Cheng, Yongkuan Wei, Juncheng |
| author_facet | Cheng, Yongkuan Wei, Juncheng |
| contents | We consider the following quasilinear Schrödinger equations of the form
\begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0, \end{equation*} where $N\geq 3,$ $p>\frac{N+2}{N-2},$ $\varepsilon>0$ and $V(x)$ is a positive function. By imposing appropriate conditions on $V(x),$ we prove that, for $\varepsilon=1,$ the existence of infinity many positive solutions with slow decaying $O(|x|^{-\frac{2}{p-1}})$ at infinity if $p>\frac{N+2}{N-2}$ and, for $\varepsilon$ sufficiently small, a positive solution with fast decaying $O(|x|^{2-N})$ if $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}.$ The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_03520 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Fast and slow decaying solutions for $H^{1}$-supercritical quasilinear Schrödinger equations Cheng, Yongkuan Wei, Juncheng Analysis of PDEs We consider the following quasilinear Schrödinger equations of the form \begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0, \end{equation*} where $N\geq 3,$ $p>\frac{N+2}{N-2},$ $\varepsilon>0$ and $V(x)$ is a positive function. By imposing appropriate conditions on $V(x),$ we prove that, for $\varepsilon=1,$ the existence of infinity many positive solutions with slow decaying $O(|x|^{-\frac{2}{p-1}})$ at infinity if $p>\frac{N+2}{N-2}$ and, for $\varepsilon$ sufficiently small, a positive solution with fast decaying $O(|x|^{2-N})$ if $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}.$ The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem. |
| title | Fast and slow decaying solutions for $H^{1}$-supercritical quasilinear Schrödinger equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1904.03520 |