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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.19917 |
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| _version_ | 1866910920740962304 |
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| author | Arkoudis, Ioannis Smyrnelis, Panayotis |
| author_facet | Arkoudis, Ioannis Smyrnelis, Panayotis |
| contents | We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions $u:\mathbb{R}^{d+k}\to\mathbb{R}^m$ of the system $Δu(x)=\nabla W(u(x))$ (with $W:\mathbb{R}^m\to \mathbb{R}$), corresponding to some nontrivial stable solutions $e:\mathbb{R}^k\to\mathbb{R}^m$. The method we propose is based on a reduction to a ground state problem in a space of functions $\mathcal H$, where $e$ is viewed as a local minimum of an effective potential defined in $\mathcal H$. As an application, by considering a heteroclinic orbit $e:\mathbb{R}\to\mathbb{R}^m$, we obtain nontrivial solutions $u:\mathbb{R}^{d+1}\to\mathbb{R}^m$ ($d\geq 2$), converging asymptotically to $e$, which can be seen as the homoclinic analogs of the heteroclinic double layers, initially constructed by Alama-Bronsard-Gui and Schatzman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground state of some variational problems in Hilbert spaces and applications to P.D.E Arkoudis, Ioannis Smyrnelis, Panayotis Analysis of PDEs 35J50. 58E99 We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions $u:\mathbb{R}^{d+k}\to\mathbb{R}^m$ of the system $Δu(x)=\nabla W(u(x))$ (with $W:\mathbb{R}^m\to \mathbb{R}$), corresponding to some nontrivial stable solutions $e:\mathbb{R}^k\to\mathbb{R}^m$. The method we propose is based on a reduction to a ground state problem in a space of functions $\mathcal H$, where $e$ is viewed as a local minimum of an effective potential defined in $\mathcal H$. As an application, by considering a heteroclinic orbit $e:\mathbb{R}\to\mathbb{R}^m$, we obtain nontrivial solutions $u:\mathbb{R}^{d+1}\to\mathbb{R}^m$ ($d\geq 2$), converging asymptotically to $e$, which can be seen as the homoclinic analogs of the heteroclinic double layers, initially constructed by Alama-Bronsard-Gui and Schatzman. |
| title | Ground state of some variational problems in Hilbert spaces and applications to P.D.E |
| topic | Analysis of PDEs 35J50. 58E99 |
| url | https://arxiv.org/abs/2504.19917 |