On positive solutions of critical semilinear equations involving the Logarithmic Laplacian
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915384482856960 |
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| author | Chen, Huyuan Zhou, Feng |
| author_facet | Chen, Huyuan Zhou, Feng |
| contents | In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_Δu= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n, \qquad\qquad\qquad\qquad\qquad\qquad$$ where $k\in(0,+\infty)$, $\mathcal{L}_Δ$ is the logarithmic Laplacian in $\mathbb{R}^n$ with $n\in\mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\tilde x,t}(x)=β_n \Big(\frac{t}{t^2+|x-\tilde x|^2)}\Big)^{\frac{n}{2}}\quad \text{ for any $t>0$, $\tilde x\in\mathbb{R}^n$},$$ where $n\in\mathbb{N}$, $β_n=2^{\frac n2} e^{\frac n2ψ(\frac n2) }>0$. When $k\in(0,+\infty)\setminus\{\frac 4n\}$, problem $(E)$ has no any positive solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04797 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On positive solutions of critical semilinear equations involving the Logarithmic Laplacian Chen, Huyuan Zhou, Feng Analysis of PDEs In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_Δu= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n, \qquad\qquad\qquad\qquad\qquad\qquad$$ where $k\in(0,+\infty)$, $\mathcal{L}_Δ$ is the logarithmic Laplacian in $\mathbb{R}^n$ with $n\in\mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\tilde x,t}(x)=β_n \Big(\frac{t}{t^2+|x-\tilde x|^2)}\Big)^{\frac{n}{2}}\quad \text{ for any $t>0$, $\tilde x\in\mathbb{R}^n$},$$ where $n\in\mathbb{N}$, $β_n=2^{\frac n2} e^{\frac n2ψ(\frac n2) }>0$. When $k\in(0,+\infty)\setminus\{\frac 4n\}$, problem $(E)$ has no any positive solution. |
| title | On positive solutions of critical semilinear equations involving the Logarithmic Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.04797 |