Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds

In this paper we continue the study on the resolvent estimates of the Laplace-Beltrami operator $\Delta_g$ on a compact manifolds $M$ with dimension $n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent $(\Delta_g+\zeta)^{-1}$ is uniformly bounded from $L^p$ to $L^q$ when $(p,q)$ are within the admissible range $p\leq2(n+1)/(n+3)$ and $q\geq2(n+1)/(n-1)$ and $\zeta$ is outside a parabola opening to the right and a small disk centered at the origin. This naturally generalizes the previous results in \cite{Kenig} and \cite{bssy} which addressed only the special case when $p=2n/(n+2), q=2n/(n-2)$. Using the shrinking spectral estimates between $L^p$ and $L^q$ we also show that when $(p,q)$ are within the interior of the admissible range, one can obtain a logarithmic improvement over the parabolic region for resolvent estimates on manifolds equipped with Riemannian metric of non-positive sectional curvature, and a power improvement depending on the exponent $(p,q)$ for flat torus. The latter therefore partially improves Shen's work in \cite{Shen} on the $L^p\to L^2$ uniform resolvent estimates on the torus. Similar to the case as proved in \cite{bssy} when $(p,q)=(2n/(n+2),2n/(n-2))$, the parabolic region is also optimal over the round sphere $S^n$ when $(p,q)$ are now in the admissible range. However, we may ask if the admissible range is sharp in the sense that it is the only possible range on the Sobolev line for which a compact manifold can have uniform resolvent estimate for $\zeta$ being ouside a parabola.Comment: A few details revise

The existence and concentration of positive ground state solutions for a class of fractional Schr\"{o}dinger-Poisson systems with steep potential wells

The present study is concerned with the following fractional Schr\"{o}dinger-Poisson system with steep potential well: \left\{% \begin{array}{ll} (-\Delta)^s u+ \la V(x)u+K(x)\phi u= f(u), & x\in\R^3, (-\Delta)^t \phi=K(x)u^2, & x\in\R^3, \end{array}% \right. where $s,t\in(0,1)$ with $4s+2t>3$, and \la>0 is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2<q<2_s^*=\frac{6}{3-2s}$, the existence of positive ground state solutions and concentration results are obtained via some new analytical skills and Nehair-Poho\v{z}aev identity. In particular, the monotonicity assumption on the nonlinearity is not necessary.Comment: 21 page

Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth

In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}% \right. where $10$ are parameters and $\Omega$ is a bounded domain in $\R^3$. We prove that there exists $\lambda_1=\lambda_1(q,\Omega)>0$ such that for any $\lambda\in(0,\lambda_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)\Delta u+u=Q(x)|u|^4u+{\lambda}f(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. for any $a,\ b>0$ and $\lambda\in \big(0,\lambda_0(q,Q,f)\big)$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between $\lambda_0$ and $b$ to show that for any (large) $\lambda>0$, there exists a $b_0(\lambda)>0$ such that the above results hold when $b>b_0(\lambda)$ and $a>0$.Comment: 25 Page

Well-posedness and ill-posedness of the 3D generalized Navier-Stokes equations in Triebel-Lizorkin spaces

In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ with $(\alpha,r)\in(1,5/4)\times[2,\infty]$ and $q_\alpha=\frac{3}{\alpha-1}$. Our work establishes a {\it dichotomy} of well-posedness and ill-posedness depending on $r=2$ or $r>2$. Specifically, by combining the new endpoint bilinear estimates in $L^{q_\alpha}_x L^2_T$ with the characterization of Triebel-Lizorkin space via fractional semigroup, we prove the well-posedness of the gNS in $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ for $r=2$. On the other hand, for any $r>2$, we show that the solution to the gNS can develop {\it norm inflation} in the sense that arbitrarily small initial data in the spaces $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ can lead the corresponding solution to become arbitrarily large after an arbitrarily short time. In particular, such dichotomy of Triebel-Lizorkin spaces is also true for the classical N-S equations, i.e.\,\,$\alpha=1$. Thus the Triebel-Lizorkin space framework naturally provides better connection between the well-known Koch-Tataru's $BMO^{-1}$ well-posed work and Bourgain-Pavlovi\'c's $\dot{B}_\infty^{-1,\infty}$ ill-posed work.Comment: 29 page

Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr\"{o}dinger equation with a potential $$iu_{t}+\Delta u-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{3}},$$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states, respectively. Our result extends the results of Hong \cite{H} and Farah-Guzm$\acute{\rm a}$n \cite{FG1} with $b\in(0,\frac12)$ to the case $0<b<1$. We then obtain a blow-up result for initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$ if $V$ satisfies some additional assumptions.0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$if$V$ satisfies some additional assumptions.Comment: arXiv admin note: text overlap with arXiv:1810.0710

Remarks on LpL^p-limiting absorption principle of Schr\"odinger operators and applications to spectral multiplier theorems

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schr\"odinger operators $H=-\Delta+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $\epsilon-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-\lambda\pm i\epsilon)^{-1}$ for all $\lambda>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schr\"odinger operators $H$. Next, we consider the fractional Schr\"odinger operator $H=(-\Delta)^{\alpha}+V$ ($0<2\alpha<n$) and prove a uniform Hardy-Littlewood-Sobolev inequality for $(-\Delta)^{\alpha}$, which generalizes the corresponding result of Kenig-Ruiz-Sogge \cite{KRS}.Comment: 15 page

Spectral multipliers, Bochner-Riesz means and uniform Sobolev inequalities for elliptic operators

This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\in {\mathbb C}\backslash [0, \infty)$, which give an extension of the second order results of Kenig-Ruiz-Sogge \cite{KRS}. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schr\"odinger operators $P(D)+V$ with small integrable potentials $V$. Finally we deduce spectral multiplier estimates for all these operators, including sharp Bochner-Riesz summability results

Decay Estimates and Strichartz Estimates of Fourth-order Schr\"{o}dinger Operator

We study time decay estimates of the fourth-order Schr\"{o}dinger operator $H=(-\Delta)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze the low energy and high energy behaviour of resolvent $R(H; z)$, and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for $e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on Jensen-Kato decay estimate and local decay estimate, we obtain the $L^1\rightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{-itH}P_{ac}$ for $d\geq5$. Furthermore, using the local decay estimate and the Georgescu-Larenas-Soffer conjugate operator method, we prove the Jensen-Kato type decay estimates for some functions of $H$.Comment: 43 Pages, published version. To appear in J. Functional Analysi

Scattering and blowup for L2L^{2}-supercritical and H˙2\dot{H}^{2}-subcritical biharmonic NLS with potentials

We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle x\rangle^\beta \big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some $\beta>N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in H^2({\bf R}^N)$ which satisfies that $$M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \ {\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}} u_{0}\|_{L^{2}}<\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}},$$ where $s_c=\frac{N}{2}-\frac{4}{p-1}\in(0,2)$, $H=\Delta^2+V$ and $Q$ is the ground state of $\Delta^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$. We crucially establish full Strichartz estimates and smoothing estimates of linear flow with a large poetential $V$, which are fundamental to our scattering results. Finally, based on the method introduced in \cite[T. Boulenger, E. Lenzmann, Blow up for biharmonic NLS, Ann. Sci. $\acute{E}$c. Norm. Sup$\acute{e}$r., 50(2017), 503-544]{B-Lenzmann}, we also prove a blow-up result for a class of potential $V$ and the radial data $u_0\in H^2({\bf R}^N)$ satisfying that $$M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \ {\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}} u_{0}\|_{L^{2}}>\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}.$$Comment: 39 page

Convex Hypersurfaces and LpL^p Estimates for Schr\"odinger Equations

This paper is concerned with Schr\"odinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the $L^p$-$L^q$ estimate of solution operator in free case. This estimate, combining with the results of fractionally integrated groups, allows us to further obtain the $L^p$ estimate of solutions for the initial data belonging to a dense subset of $L^p$ in the case of integrable potentials.Comment: 18 page