Hongjie Dong, Yanze Liu
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $α\in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.
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In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.
Weighted estimates for time-fractional parabolic equations with VMO coefficients
本文研究了具VMO系数的时间分数阶抛物方程在全空间和半空间中的加权估计与可解性，通过奇偶延拓及边界条件推导半空间结果，证明依赖于Fefferman-Stein定理与Hardy-Littlewood极大函数定理。
Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative problems for parabolic equations defined in a half-space.
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher Hölder regularity for such nonlocal equations.
An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on $t$. The method of proving simplifies the methods previously used in the case $p=q$.
We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli's type estimates for the Stokes systems with variable coefficients. A new $ε$-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regular…
On parabolic equations in Morrey spaces with VMO $a$ and Morrey $b,c$
本文研究了全空间中具有 VMO $a$ 和 Morrey $b,c$ 系数的二阶抛物方程，证明了其解在混合范数 Morrey 空间中的存在性与唯一性。
We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time d…
The unique solvability of parabolic equations in Sobolev spaces with mixed norms is presented. The second order coefficients (except $a^{11}$) are assumed to be only measurable in time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in one spatial variable and VMO in the other variables.
We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet bou…
The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and be almost in VMO (vanishing mean oscillation) with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic Itô equations in the class of "good" solutions (which is nonempty). This also implies uniqueness of a Green's function in the class of "good" ones (which is always nonempty)…
We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.
We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(Ω)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{α,p}$-regularity for $α> \frac{1}{2}$ in time.
We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations.
Global Calderón-Zygmund theory for fractional Laplacian type equations
本文研究非齐次 $s$-分数阶 Laplace 型方程的边界正则性，依据核系数的 VMO、Dini 或 Hölder 条件，证明 $u/d^s$ 的尖锐全局 Calderón-Zygmund 估计。
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-s…
We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\m…
Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3<p,q<\infty$ and $u_h\in L^{\infty}(0,T; BMO^{-1}(\R^3))$ with $u_h(T)\in VMO^{-1}(\R^3)$, then $u$ can be extended beyond $T$. This result generalizes the recent result proved by Gallagher, Koch and Planchon, which requires $u\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$. Our proof is based on a new interior regularity criterion in terms of one velocity component.
We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable $v$ and satisfy a vanishing mean oscillation (VMO) type condition. We consider the $L_2$ case separately and treat more general equations which include the relativistic KFP equation.
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse Hölder's inequalities are established. Lipschitz estimates for weak solutions are proved for homogeneous equations with singular degenerate coefficients depending only on one spatial variable. These estimates are then used to establish interior, boundary, and global weighted estimates of Calderón-Zygmund type for weak solutions, assuming that the coefficients are partially VMO (vani…
We establish some new properties of the Dunkl-Wiener amalgam spaces defined on the real line. These results allow us to obtain the boundedness of Dunkl-type fractional integral and fractional maximal operators in the Dunkl-Fofana spaces.
We show that the symmetric radial decreasing rearrangement can increase the fractional Gagliardo semi-norm in domains.
When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on the whole domain? On what do the comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove some misconceptions by counter-examples.
We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.
We investigate the existence of solutions to the fractional nonlinear Schrödinger equation $(-Δ)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.
Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content
本文研究了非中心分数极大函数在关联于二进Hausdorff内容的平均振荡空间上的作用，改进了其在BMO和VMO函数上的现有结果，并证明了多维非中心极大函数在相应空间上的有界性。
The purpose of this paper is to establish the existence of solutions with prescribed norm to a class of nonlinear equations involving the mixed fractional Laplacians. This type of equations arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. Our method is novel and our results cover all the previous ones.
Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on $\R^2$ and $\R^3$
利用倒置I-方法和更高阶修正能量方法，证明了二维和三维空间中具有三次和Hartree型非线性项的散焦分数阶非线性薛定谔方程解的高Sobolev范数的多项式增长界。
Uniform estimates for elliptic systems with critical Sobolev coefficients
本研究针对系数属于临界Sobolev-Lorentz空间的散度型椭圆方程组，建立了由系数范数和椭圆性常数控制的一致估计，解决了经典理论无法控制估计常数的问题。
On the almost sure growth of Hölder norms for the 1d periodic fractional BBM equation
利用Tzvetkov的拟不变性策略和Bourgain的整体化论证，将L²确定性控制几乎必然扩展到L∞框架，得到解的Hölder范数的几乎必然多项式界。
Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations
研究二次导数分数阶非线性薛定谔方程在Sobolev空间中的适定性，通过迭代项估计证明范数膨胀和无限正则性损失，从而得出不适定性。
Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes
本文研究生物膜斑图形成的一维变分模型，推导最小能量标度律，并提出用Cahn-Hillard/Modica-Mortola能量界定分数阶Sobolev半范数的新非线性插值不等式作为主要工具。
We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(Ω)}+\int_ΩW(u)\,dx, $$ where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\mathbb{R}^n$. The results collected here will also be useful for forthcoming papers, where the second and the third author will study the $Γ$-convergence and the density estimates for level sets of minimizers.
We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ with respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u=λu+μ|u|^{q-2}u+|u|^{p-2}u \quad \hbox{in $\mathbb{R}^3$,} \end{equation*} with a prescribed mass \begin{equation*} \int_{\mathbb{R}^{3}}|u|^{2}dx=c^{2}, \end{equation*} where $s\in(0, 1)$, $a, b, c>0$, $2<q<p<2_{s}^{\ast}=\frac{6}{3-2s}$, $μ>0$ and $λ\in\mathbb{R}$ as a Lagrange multiplier. Under different assumptions on $q<p$, $c>0$ and $μ>0$, we prove some existence results about…
Modified Wasserstein gradient flow formulation of time-fractional porous medium equations with nonlocal pressure
针对带非局部压力的时间分数阶多孔介质方程，提出基于修正Wasserstein距离的JKO格式和平方分数阶Sobolev范数，证明弱解存在性并建立正则化效应与Lp范数估计。
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_{s}^{\ast}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $μ\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we firs…
We consider the following coupled fractional Schrödinger system: \begin{equation*} \left\{ \begin{aligned} &(-Δ)^su+λ_1u=μ_1|u|^{2p-2}u+β|v|^p|u|^{p-2}u\\ &(-Δ)^sv+λ_2v=μ_2|v|^{2p-2}v+β|u|^p|v|^{p-2}v\\ \end{aligned} \right.\quad\text{in}~{\mathbb{R}^N}, \end{equation*} with $0<s<1$, $2s<N\le 4s$ and $1+\frac{2s}{N}<p<\frac{N}{N-2s}$, under the following constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2dx=a_1^2\quad\text{and}\quad \int_{\mathbb{R}^N}|v|^2dx=a_2^2. \end{align*} Assuming that the parameters $μ_1,μ_2,a_1, a_2$ are fixed quantities, we prove the existence of normalized solution f…
We consider a family of fractional porous media equations, recently studied by Caffarelli and Vázquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.
We obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation $u_t+(-Δ)^{σ/2}u^m=0$, posed in the whole space with $0<σ<2$, $0<m\le 1$. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the $L^1$-norm and the Marcinkiewicz norm. The estimates contain exact exponents and best constants. We also obtain optimal estimates for the extinction time of the solutions in the range $m$ near 0 where solutions may vanish completely in finite time. Actually, our results apply to equations with a more general…
Polynomial growth of Sobolev norms of solutions of the fractional NLS equation on $\mathbb{T}^d$
本文证明了分数非线性Schrödinger方程在\T^d上解的Sobolev范数的多项式增长界，通过建立分数Schrödinger方程在\T^d上的Strichartz估计来克服高维中的非均匀性问题。
Normalized solutions for fractional Choquard equation with critical growth on bounded domain
本文利用极小化技术和一致山路定理，证明了星形有界域上临界分数阶Choquard方程正解的多重性，分别得到第一解和第二解。
Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials
本文研究带势分数阶 Schrödinger 方程在 $L^2$ 范数约束下的规范化解，利用 $h$ 的全局极大值点证明当 $\varepsilon$ 足够小时解具有相应多重性。
In this article we prove both norm and modular Hardy inequalities for a class functions in one-dimensional fractional Orlicz-Sobolev spaces.
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide short proofs of the Frank-Seiringer "Hardy inequalities" where fractional smoothness is measured by a Besov norm.
In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians $$ (-Δ)^{s_1} u +(-Δ)^{s_2} u + λu=|u|^{p-2} u $$ under the constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<s_2<s_1<1$, $2+ \frac {4s_1}{N} \leq p< \infty $ if $N \leq 2s_1$, $2+ \frac {4s_1}{N} \leq p<\frac{2N}{N-2s_1}$ if $N >2s_1$, $λ\in \R$ appearing as Lagrange multiplier is unknown. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^N$, where $\mathcal{F}$ denotes the Fourier…
Weighted mixed-norm estimates for fractional parabolic equations with space-time nonlocal operators
该研究针对带时空非局部算子的分数阶抛物方程，建立加权混合范数估计，推导奇混合范数空间对应结果，证明其鲁棒性与唯一可解性，采用逐方向延拓论证完成证明。
In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator.
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to proving a fractional version of the classi…
In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of $\mathcal{\dot{C}}^η$, for $η\in(0,1)$.
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a norm on $\rn$ with unit ball $K$ are shown to converge to the anisotropic Sobolev seminorm of $f$ defined by the norm with unit ball $\,\ompd K$, the polar $L_p$ moment body of $K$. The limiting behavior for $s\to 0^+$ is also determined (extending results by Maz$'$ya & Shaposhnikova).
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\mathbb{R}^n)$ converges to the Sobolev seminorm of $f$ as $s\rightarrow1^-$. Ludwig introduced the anisotropic fractional Sobolev $s$-seminorms of $f$ defined by a norm on $\mathbb{R}^n$ with unit ball $K$, and showed that they converge to the anisotropic Sobolev seminorm of $f$ defined by the norm whose unit ball is the polar $L_p$ moment body of $K$, as $s\rightarrow1^-$. The asymmetric anisotropic $s$-seminorms are shown to converge to the anisotropic Sobolev sem…
We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case w…
The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.
We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert \mathbf{G} ν\Vert_{L^q(Ω, d σ)} \le C \, \Vert ν\Vert, \quad \text{for all positive measures $ν$ in $Ω$},$ along with their weak-type counterparts, where $\Vert ν\Vert=ν(Ω)$, and $G$ is an integral operator with nonnegative kernel, $\mathbf{G} ν(x) = \int_ΩG(x, y) d ν(y).$ These problems are motivated by sublinear elliptic equations in a domain $Ω\subset\mathbb{R}^n$ with non-trivial Green's function $G(x, y)$ associated with the Laplacian, fractional Laplacian, or more general elliptic operator. We also treat fractional …
We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier--Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomp…
In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schrödinger equation \begin{equation} i\partial_{t}ψ=(-Δ)^{s}ψ-f(ψ), \qquad (0.1)\end{equation} where $0<s<1$, $f(ψ)=|ψ|^{p}ψ$ with $\frac{4s}{N}<p<\frac{4s}{N-2s}$ or $f(ψ)=(|x|^{-γ}\ast|ψ|^2)ψ$ with $2s<γ<\min\{N,4s\}$. To this end, we look for normalized solutions of the associated stationary equation \begin{equation} (-Δ)^s u+ωu-f(u)=0. \qquad (0.2) \end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized sol…
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