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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.07577 |
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| _version_ | 1866910995932250112 |
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| author | Lanz, Florian P. Lenzmann, Enno |
| author_facet | Lanz, Florian P. Lenzmann, Enno |
| contents | We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation
$$
(-Δ)^s u = e^u \quad \mbox{in $\mathbb{R}$} \quad \mbox{with} \quad \int_{\mathbb{R}} e^u dx < +\infty
$$ for all exponents $s \in (\frac{1}{2},1)$. Furthermore, we show $u$ has finite Morse index and that its linearized operator is nondegenerate. Our arguments are based on a fixed point scheme in terms of the function $v= \sqrt{e^u}$ and we devise a nonlocal shooting method involving (locally) compact nonlinear maps. We also study existence, symmetry and uniqueness of solutions to $(-Δ)^s u = K e^u$ in $\mathbb{R}$ with $K e^u \in L^1(\mathbb{R})$ for a general class of positive, even and monotone-decreasing functions $K > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$ Lanz, Florian P. Lenzmann, Enno Analysis of PDEs We prove existence, symmetry and uniqueness of solutions to the fractional Gelfand equation $$ (-Δ)^s u = e^u \quad \mbox{in $\mathbb{R}$} \quad \mbox{with} \quad \int_{\mathbb{R}} e^u dx < +\infty $$ for all exponents $s \in (\frac{1}{2},1)$. Furthermore, we show $u$ has finite Morse index and that its linearized operator is nondegenerate. Our arguments are based on a fixed point scheme in terms of the function $v= \sqrt{e^u}$ and we devise a nonlocal shooting method involving (locally) compact nonlinear maps. We also study existence, symmetry and uniqueness of solutions to $(-Δ)^s u = K e^u$ in $\mathbb{R}$ with $K e^u \in L^1(\mathbb{R})$ for a general class of positive, even and monotone-decreasing functions $K > 0$. |
| title | Existence and Uniqueness for the Fractional Gelfand Equation in $\mathbb{R}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.07577 |