Advances in Nonlinear Analysis (Mar 2025)
Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
Abstract
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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