Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913463993892864 |
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| author | Kumagai, Kenta |
| author_facet | Kumagai, Kenta |
| contents | We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $λ$. The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $λ=0$. As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in $N$ dimensions with $3\le N \le 9$ and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $2\le N\le 9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions Kumagai, Kenta Analysis of PDEs 35B32, 35J61, 35J25, 35B35 We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $λ$. The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $λ=0$. As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in $N$ dimensions with $3\le N \le 9$ and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $2\le N\le 9$. |
| title | Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions |
| topic | Analysis of PDEs 35B32, 35J61, 35J25, 35B35 |
| url | https://arxiv.org/abs/2404.04019 |