Ground state solutions to the nonlinear Born-Infeld problem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918260350386176 |
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| author | Bieganowski, Bartosz Ikoma, Norihisa Mederski, Jarosław |
| author_facet | Bieganowski, Bartosz Ikoma, Norihisa Mederski, Jarosław |
| contents | In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground state solutions to the nonlinear Born-Infeld problem Bieganowski, Bartosz Ikoma, Norihisa Mederski, Jarosław Analysis of PDEs 35A15, 35J25, 35J93, 35Q75 In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$. |
| title | Ground state solutions to the nonlinear Born-Infeld problem |
| topic | Analysis of PDEs 35A15, 35J25, 35J93, 35Q75 |
| url | https://arxiv.org/abs/2512.19566 |