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Register a new userFourth order Schrodinger equation with mixed dispersion on Cartan-Hadamard manifolds
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We study the fourth order Schrodinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.
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We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr{o}dinger equation begin{equation}label{1} {Delta}^{2}u+mu Delta u-{lambda}u={|u|}^{p-2}u, ~~~~x in R^{N} end{equation} under the normalized constraint $$int_{{mathbb{R}^N}} {{u}^2}=a^2,$$ where $N!geq!2$, $a,mu!>!0$, $2+frac{8}{N}!<!p!<! 4^{*}!=!frac{2N}{(N-4)^{+}}$ and $lambdainR$ appears as a Lagrange multiplier. Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{mu}(u)=frac{1}{2}{||Delta u||}_2^2-frac{mu}{2}{|| abla u||}_2^2-frac{1}{p}{||u||}_p^p $$ we could find at least two normalized solutions to (ref{1}) if $2!+!frac{8}{N}!<! p!<!{ 4^{*} }$ and $mu^{pgamma_p-2}a^{p-2}!<!C$ for some explicit constant $C!=!C(N,p)!>!0$ and $gamma_p!=!frac{N(p!-!2)}{4p}$. Furthermore, we give some asymptotic properties of the normalized solutions to (ref{1}) as $muto0^+$ and $ato0^+$, respectively. In conclusion, we mainly extend the results in cite{DBon,dbJB}, which deal with (ref{1}), from $muleq0$ to the case of $mu>0$, and also extend the results in cite{TJLu,Nbal}, which deal with (ref{1}), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.
We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.
We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincare inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Greens function vanishing at infinity. On the source function we assume a sharp pointwise decay depending on the weight appearing in the Poincare inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold.
We prove global weighted Strichartz estimates for radial solutions of linear Schrodinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity.
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Gauduchon which extends the Calabi conjecture; this was one of the original motivations to this work. We were also motivated by the fact that there had been increasing interests in fully nonlinear pdes from complex geometry in recent years, and aimed to develop general methods to cover as wide a class of equations as possible.
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