Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910786243264512 |
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| author | Soave, Nicola |
| author_facet | Soave, Nicola |
| contents | We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $μ|u|^{q-2} u$ we prove several existence/non-existence and stability/instability results.
This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_02003 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case Soave, Nicola Analysis of PDEs We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $μ|u|^{q-2} u$ we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$. |
| title | Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1901.02003 |