Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
In this article, we investigate the following Schrödinger equation: −Δu=h(x)g(u)+λuinRN,∫RN∣u∣2dx=au∈H1(RN),\left\{\begin{array}{ll}-\Delta u=h\left(x)g\left(u)+\lambda u\hspace{1.0em}& \hspace{-0.2em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-04-01
|
| Series: | Demonstratio Mathematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/dema-2025-0113 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In this article, we investigate the following Schrödinger equation: −Δu=h(x)g(u)+λuinRN,∫RN∣u∣2dx=au∈H1(RN),\left\{\begin{array}{ll}-\Delta u=h\left(x)g\left(u)+\lambda u\hspace{1.0em}& \hspace{-0.2em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}\hspace{0.1em}\text{d}\hspace{0.1em}x=a\hspace{1.0em}& u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where N≥3N\ge 3, a>0a\gt 0, and λ∈R\lambda \in {\mathbb{R}} arises as a Lagrange multiplier. Assuming that the sublinear nonlinearity f∈C(RN×R,R)f\in C\left({{\mathbb{R}}}^{N}\times {\mathbb{R}},{\mathbb{R}}) is locally defined for ∣u∣| u| small; by using generalized minimax approach, we prove the existence of infinitely many normalized solutions {(uk,λk)}k∈N{\{\left({u}_{k},{\lambda }_{k})\}}_{k\in {\mathbb{N}}} for the problem, where λk<0{\lambda }_{k}\lt 0 and ∣uk(x)∣→0| {u}_{k}\left(x)| \to 0 as ∣x∣→∞| x| \to \infty . Furthermore, as k→∞k\to \infty , we show that ‖∇uk‖L2(RN)→0{\Vert \nabla {u}_{k}\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{N})}\to 0, ‖uk‖L∞(RN)→0{\Vert {u}_{k}\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{N})}\to 0, and λk→0{\lambda }_{k}\to 0. |
|---|---|
| ISSN: | 2391-4661 |