A generalised Nehari manifold method for a class of non linear Schrödinger systems in $\mathbb{R}^3$
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| Format: | Preprint |
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2024
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| author | Cortopassi, Tommaso Georgiev, Vladimir |
| author_facet | Cortopassi, Tommaso Georgiev, Vladimir |
| contents | We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is $-ε^2 Δu + V(x) u= h_v(u,v), - ε^2 Δv + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_ε,v_ε)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $ε\to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_18483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalised Nehari manifold method for a class of non linear Schrödinger systems in $\mathbb{R}^3$ Cortopassi, Tommaso Georgiev, Vladimir Analysis of PDEs 35Q55, 35Q51 We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is $-ε^2 Δu + V(x) u= h_v(u,v), - ε^2 Δv + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_ε,v_ε)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $ε\to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$. |
| title | A generalised Nehari manifold method for a class of non linear Schrödinger systems in $\mathbb{R}^3$ |
| topic | Analysis of PDEs 35Q55, 35Q51 |
| url | https://arxiv.org/abs/2402.18483 |