Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity
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2024
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| author | Zhang, He Chen, Haibo Yao, Shuai Sun, Juntao |
| author_facet | Zhang, He Chen, Haibo Yao, Shuai Sun, Juntao |
| contents | We investigate normalized solutions for a class of nonlinear Schrödinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+β|u|^{p-2}u$ on a bounded domain $Ω$. Firstly, when $2+\frac{4}{N}<q<p\leq2^*:=\frac{2N}{N-2}$ and $β=-1$, under an explicit smallness assumption on $V$, we prove the existence of a global minimum solution and a high-energy solution if the mass is large enough. For this case we do not require that $Ω$ is star-shaped, which partly solves an open problem by Bartsch et al. [Math. Ann. 390 (2024) 4813--4859]. Moreover, we find that the global minimizer also exists although the nonlinearity is $L^2$-supercritical. Secondly, when $2<q<2+\frac{4}{N}<p=2^*$ and $β=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $Ω$ is star-shaped and the mass is small enough. It seems to be new in the study of normalized ground state in the context of the Brézis-Nirenberg problem, even for the autonomous case of $V(x)\equiv0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17951 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity Zhang, He Chen, Haibo Yao, Shuai Sun, Juntao Analysis of PDEs 35J20, 35J60, 35Q55 We investigate normalized solutions for a class of nonlinear Schrödinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+β|u|^{p-2}u$ on a bounded domain $Ω$. Firstly, when $2+\frac{4}{N}<q<p\leq2^*:=\frac{2N}{N-2}$ and $β=-1$, under an explicit smallness assumption on $V$, we prove the existence of a global minimum solution and a high-energy solution if the mass is large enough. For this case we do not require that $Ω$ is star-shaped, which partly solves an open problem by Bartsch et al. [Math. Ann. 390 (2024) 4813--4859]. Moreover, we find that the global minimizer also exists although the nonlinearity is $L^2$-supercritical. Secondly, when $2<q<2+\frac{4}{N}<p=2^*$ and $β=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $Ω$ is star-shaped and the mass is small enough. It seems to be new in the study of normalized ground state in the context of the Brézis-Nirenberg problem, even for the autonomous case of $V(x)\equiv0$. |
| title | Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity |
| topic | Analysis of PDEs 35J20, 35J60, 35Q55 |
| url | https://arxiv.org/abs/2411.17951 |