Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
The present study investigates the existence and non-degeneracy of normalized solutions for the following fractional Schrödinger equation: (−Δ)su+V(x)u=aup+μu,x∈RN,u∈Hs(RN){\left(-\Delta )}^{s}u+V\left(x)u=a{u}^{p}+\mu u,\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}u\in {H}^{s}\left({{\mathbb...
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2025-03-01
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| author | Guo Qing Zhang Yuhang |
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| description | The present study investigates the existence and non-degeneracy of normalized solutions for the following fractional Schrödinger equation: (−Δ)su+V(x)u=aup+μu,x∈RN,u∈Hs(RN){\left(-\Delta )}^{s}u+V\left(x)u=a{u}^{p}+\mu u,\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}u\in {H}^{s}\left({{\mathbb{R}}}^{N}) with the L2{L}^{2}-restriction ∫RNu2(x)dx=1{\int }_{{{\mathbb{R}}}^{N}}{u}^{2}\left(x){\rm{d}}x=1, where s∈(0,1)s\in \left(0,1), p∈(1,2NN−2s−1)p\in \left(1,\frac{2N}{N-2s}-1), N>2sN\gt 2s, a>0a\gt 0 and V(x)V\left(x) is some smooth trapping potential. Via a Lyapunov-Schmidt variational reduction, we first construct solutions of the form ua∼−μaa1p−1∑j=1kU(−μa)12s(x−xa,j),{u}_{a} \sim {\left(\frac{-{\mu }_{a}}{a}\right)}^{\tfrac{1}{p-1}}\mathop{\sum }\limits_{j=1}^{k}U\left(\phantom{\rule[-0.75em]{}{0ex}},{\left(-{\mu }_{a})}^{\tfrac{1}{2s}}\left(x-{x}_{a,j})\right), where xa,j{x}_{a,j} approach suitable critical points of V(x)V\left(x), U(x)∈Hs(RN)U\left(x)\in {H}^{s}\left({{\mathbb{R}}}^{N}) is the unique radially symmetric positive ground state solution of (−Δ)su+u=up,u(0)=maxx∈RNu(x){\left(-\Delta )}^{s}u+u={u}^{p},u\left(0)={\max }_{x\in {{\mathbb{R}}}^{N}}u\left(x). Subsequently, the local Pohozaev identity techniques are applied to establish the non-degeneracy of such normalized solutions. This study successfully addresses the complexities arising from the non-local characteristics of the fractional Laplacian in the local analysis and pointwise estimates of solutions. In contrast to the unconstrained scenario, the mass-critical power, denoted as p=4sN+1p=\frac{4s}{N}+1, acts as a pivotal threshold. It delineates distinct ranges of values for pp, each corresponding to vastly different concentration behaviors exhibited by the solutions. This phenomenon unequivocally underscores the profound impact of constraint conditions on the intricate dynamics of the solutions. |
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| spelling | doaj-art-dc43d99b2858496d80ebddfee8bd12692025-08-20T02:12:10ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-03-011416133616210.1515/anona-2025-0069Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equationsGuo Qing0Zhang Yuhang1College of Science, Minzu University of China, Beijing 100081, ChinaCollege of Science, Minzu University of China, Beijing 100081, ChinaThe present study investigates the existence and non-degeneracy of normalized solutions for the following fractional Schrödinger equation: (−Δ)su+V(x)u=aup+μu,x∈RN,u∈Hs(RN){\left(-\Delta )}^{s}u+V\left(x)u=a{u}^{p}+\mu u,\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}u\in {H}^{s}\left({{\mathbb{R}}}^{N}) with the L2{L}^{2}-restriction ∫RNu2(x)dx=1{\int }_{{{\mathbb{R}}}^{N}}{u}^{2}\left(x){\rm{d}}x=1, where s∈(0,1)s\in \left(0,1), p∈(1,2NN−2s−1)p\in \left(1,\frac{2N}{N-2s}-1), N>2sN\gt 2s, a>0a\gt 0 and V(x)V\left(x) is some smooth trapping potential. Via a Lyapunov-Schmidt variational reduction, we first construct solutions of the form ua∼−μaa1p−1∑j=1kU(−μa)12s(x−xa,j),{u}_{a} \sim {\left(\frac{-{\mu }_{a}}{a}\right)}^{\tfrac{1}{p-1}}\mathop{\sum }\limits_{j=1}^{k}U\left(\phantom{\rule[-0.75em]{}{0ex}},{\left(-{\mu }_{a})}^{\tfrac{1}{2s}}\left(x-{x}_{a,j})\right), where xa,j{x}_{a,j} approach suitable critical points of V(x)V\left(x), U(x)∈Hs(RN)U\left(x)\in {H}^{s}\left({{\mathbb{R}}}^{N}) is the unique radially symmetric positive ground state solution of (−Δ)su+u=up,u(0)=maxx∈RNu(x){\left(-\Delta )}^{s}u+u={u}^{p},u\left(0)={\max }_{x\in {{\mathbb{R}}}^{N}}u\left(x). Subsequently, the local Pohozaev identity techniques are applied to establish the non-degeneracy of such normalized solutions. This study successfully addresses the complexities arising from the non-local characteristics of the fractional Laplacian in the local analysis and pointwise estimates of solutions. In contrast to the unconstrained scenario, the mass-critical power, denoted as p=4sN+1p=\frac{4s}{N}+1, acts as a pivotal threshold. It delineates distinct ranges of values for pp, each corresponding to vastly different concentration behaviors exhibited by the solutions. This phenomenon unequivocally underscores the profound impact of constraint conditions on the intricate dynamics of the solutions.https://doi.org/10.1515/anona-2025-0069fractional schrödiner equationsnormalized solutionsreduction methodpohozaev identities35b2535b4435b09 |
| spellingShingle | Guo Qing Zhang Yuhang Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations Advances in Nonlinear Analysis fractional schrödiner equations normalized solutions reduction method pohozaev identities 35b25 35b44 35b09 |
| title | Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations |
| title_full | Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations |
| title_fullStr | Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations |
| title_full_unstemmed | Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations |
| title_short | Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations |
| title_sort | existence and non degeneracy of the normalized spike solutions to the fractional schrodinger equations |
| topic | fractional schrödiner equations normalized solutions reduction method pohozaev identities 35b25 35b44 35b09 |
| url | https://doi.org/10.1515/anona-2025-0069 |
| work_keys_str_mv | AT guoqing existenceandnondegeneracyofthenormalizedspikesolutionstothefractionalschrodingerequations AT zhangyuhang existenceandnondegeneracyofthenormalizedspikesolutionstothefractionalschrodingerequations |