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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.03994 |
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| _version_ | 1866911760703815680 |
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| author | Dai, Wei Duan, Lixiu Zhang, Rong |
| author_facet | Dai, Wei Duan, Lixiu Zhang, Rong |
| contents | In this paper, without any assumption on $v$ and under the extremely mild assumption $u(x)= O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large, we classify solutions of the following conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{3}$: $$ \begin{cases} \ (-Δ)^{\frac{1}{2}} u=v^{4} ,&x\in \mathbb{R}^{3},\\ \ -Δv=e^{pw} ,&x\in \mathbb{R}^{3},\\ \ (-Δ)^{\frac{3}{2}} w=u^{3} ,&x\in \mathbb{R}^{3}, \end{cases} $$ where $p>0$, $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{\mathbb{R}^{3}}u^{3}(x)\mathrm{d}x<+\infty$. Moreover, under the extremely mild assumption that \emph{either} $u(x)$ or $v(x)=O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large \emph{or} $\int_{\mathbb{R}^{4}}e^{Λpw(y)}\mathrm{d}y<+\infty$ for some $Λ\geq1$, we also prove classification of solutions to the conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{4}$: \begin{align*} \begin{cases} \ (-Δ)^{\frac{1}{2}} u=e^{pw} ,&x\in \mathbb{R}^{4},\\ \ -Δv=u^2 ,&x\in \mathbb{R}^{4},\\ \ (-Δ)^{2} w=v^{4} ,&x\in \mathbb{R}^{4}, \end{cases} \end{align*} where $p>0$, and $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{\mathbb{R}^{4}}v^{4}(x)\mathrm{d}x<+\infty$. The key ingredients are deriving the integral representation formulae and crucial asymptotic behaviors of solutions $(u,v,w)$ and calculating the explicit value of the total curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_03994 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of solutions to $3$-D and $4$-D mixed order conformally invariant systems with critical and exponential growth Dai, Wei Duan, Lixiu Zhang, Rong Analysis of PDEs Primary: 35M30, Secondary: 35A02, 53C18, 35R11 In this paper, without any assumption on $v$ and under the extremely mild assumption $u(x)= O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large, we classify solutions of the following conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{3}$: $$ \begin{cases} \ (-Δ)^{\frac{1}{2}} u=v^{4} ,&x\in \mathbb{R}^{3},\\ \ -Δv=e^{pw} ,&x\in \mathbb{R}^{3},\\ \ (-Δ)^{\frac{3}{2}} w=u^{3} ,&x\in \mathbb{R}^{3}, \end{cases} $$ where $p>0$, $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{\mathbb{R}^{3}}u^{3}(x)\mathrm{d}x<+\infty$. Moreover, under the extremely mild assumption that \emph{either} $u(x)$ or $v(x)=O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large \emph{or} $\int_{\mathbb{R}^{4}}e^{Λpw(y)}\mathrm{d}y<+\infty$ for some $Λ\geq1$, we also prove classification of solutions to the conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{4}$: \begin{align*} \begin{cases} \ (-Δ)^{\frac{1}{2}} u=e^{pw} ,&x\in \mathbb{R}^{4},\\ \ -Δv=u^2 ,&x\in \mathbb{R}^{4},\\ \ (-Δ)^{2} w=v^{4} ,&x\in \mathbb{R}^{4}, \end{cases} \end{align*} where $p>0$, and $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total curvature condition $\int_{\mathbb{R}^{4}}v^{4}(x)\mathrm{d}x<+\infty$. The key ingredients are deriving the integral representation formulae and crucial asymptotic behaviors of solutions $(u,v,w)$ and calculating the explicit value of the total curvature. |
| title | Classification of solutions to $3$-D and $4$-D mixed order conformally invariant systems with critical and exponential growth |
| topic | Analysis of PDEs Primary: 35M30, Secondary: 35A02, 53C18, 35R11 |
| url | https://arxiv.org/abs/2401.03994 |