We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $μ|u|^{q-2} u$ we prove several existence/non-existence and stability/instability results.
This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$.
We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.
We investigate the existence of ground states for the focusing subcritical NLS energy on metric graphs with localized nonlinearities. In particular, we find two thresholds on the measure of the region where the nonlinearity is localized that imply, respectively, existence or nonexistence of ground states. In order to obtain these results we adapt to the context of metric graphs some classical techniques from the Calculus of Variations.
In this note we study analytically and numerically the existence and stability of standing waves for one dimensional nonlinear Schrödinger equations whose nonlinearities are the sum of three powers. Special attention is paid to the curves of non-existence and curves of stability change on the parameter planes.
We discuss a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs.
We consider the energy critical nonlinear Schrodinger equation on generic irrational tori. Using the long-time Strichartz estimates proved in [8], we establish polynomial upper bounds for higher Sobolev norms for solutions with small energy.
This paper proves existence and stability results of solitary-wave solutions to coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of solitary waves is obtained by solving a variational problem subject to two independent constraints and using the concentration-compactness method. The set of minimizers is shown to be stable and further information about the structures of this set are given. The paper extends the results previously obtained by Cipolatti and Zumpichiatti, Nguyen and Wang, and Ohta.
We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrödinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k, the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.
A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…
We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) …
We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the $2d$ cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, $k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.
We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.
We study the quintic Non Linear Schrödinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.
The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations.
Global existence and scattering for the nonlinear defocusing Schrödinger equation in 2 dimensions are known for domains exterior to star-shaped obstacles and for nonlinearities that grow at least as the quintic power. In this paper, we extend the global existence result for all non-trapping obstacles and for nonlinearities with power strictly greater than quartic. For such nonlinearities, we also prove scattering for a class of so-called almost star-shaped obstacles.
We consider the cubic Nonlinear Schrodinger Equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time H^s bounds in terms of the H^s size of the initial data for s greater than or equal to -1/6.
In this paper we consider Schr{ö}dinger equations with nonlinearities of odd order 2$σ$ + 1 on T^d. We prove that for $σ$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{ö}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3.
All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least $C^\infty$) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the first result on exis…
We prove the existence of quasi-periodic solutions for Schroedinger equations with a multiplicative potential on T^d, d \geq 1, merely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. …
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|…
We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subc…
We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.
In this paper, we are concerned with the following type of elliptic problems: $$ (-Δ)^α u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^α({\mathbb R}^N), $$ where $2<q< 2^*$, $0<α<1$, $0<s<2α$, $2^*_{s}=2(N-s)/(N-2α)$ is the critical Sobolev-Hardy exponent, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to the problem, we obtain the existence of positive solutions under certain assumptions on $a(x),k(x)$.
We consider a completely resonant nonlinear Schrödinger equation on the $d$-dimensional torus, for any $d\geq 1$, with polynomial nonlinearity of any degree $2p+1$, $p\geq1$, which is gauge and translation invariant. We study the behaviour of high Sobolev $H^{s}$-norms of solutions, $s\geq s_1+1 > d/2 + 2$, whose initial datum $u_0\in H^{s}$ satisfies an appropriate smallness condition on its low $H^{s_1}$ and $L^2$-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm $H^{s}$ over finite but long time scale that is exponential in the regularity param…
The L^2 -critical defocusing nonlinear Schrodinger initial value problem on R^d is known to be locally well-posed for initial data in L^2. Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data u_0 in Sobolev H^1 and for data in the weighted space (1+|x|) u_0 in L^2. For the d=2 problem, it is known that global existence holds for data in H^s and also for data in the weighted space (1+|x|)^σ u_0 in L^2 for certain s, σ< 1. We prove: If global well-posedness holds in H^s then global existence and scattering holds for initial data i…
This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} Δ^{2}u-λu=μ|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $μ>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $λ$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are p…
We consider the completely resonant defocusing non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix $s>1$. We show the existence of solutions of this equation which achieve arbitrarily large growth of $H^s$ Sobolev norms. We also give estimates for the time required to attain this growth.
We consider the Derivative NLS equation with general quadratic nonlinearities. In \cite{be2} the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$. Here we prove a similar result for large initial data in all dimensions $n \geq 2$.
Nonlinear Schrödinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically.
Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schrödinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic nonlinearities, and get polynomial in time bounds for any higher order nonlinearities. In $3d$, we prove that solutions to the cubic NLS grow at most exponentially, while for sub-cubic NLS we get polynomial bounds on the growth of the $H^2$-norm.
We consider a class of fully nonlinear Schrödinger equations and we prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions. We deal with reversible autonomous nonlinearities and we look for analytic solutions. Note that this is the first result on analytic quasi-periodic solutions for fully nonlinear PDEs.
We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces $H^s(\mathbb{T})$, $s > -\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in $H^s(\mathbb{T})$, $s > -\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo-Oh for the cubic NLS, this also leads to non-existence of solutions for the (non-renormalized) 4NLS in negative Sobo…
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on $\RN$, when $N\geq2$, \begin{equation} \Lp u=λ\hspace{0.2mm}K(x)u^{r-1}-V(x)u^{q-1}.\nonumber \end{equation} Here, $K(x),V(x)>0$ are suitable potentials, $1<p<r<q<\infty$, and $λ>0$ is a parameter. To study this problem, some compact embedding results regarding $\MVR\hookrightarrow\LKR$ are proved that unify and extend some recent results of the author \cite{Ha1,Ha2,Ha3}.
In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<σ< σ_{c}(s)$ and the initial data is sufficiently small, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and…
In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities, \begin{align*} (-Δ_p)^su - μ\dfrac{\vert u \vert^{p-2} u}{\vert x \vert^{ps}} = \dfrac{\vert u\vert^{p^*_s(β)-2}u}{\vert x \vert^β} + \dfrac{\vert u\vert^{p^*_s(α)-2} u}{\vert x \vert^α}, \end{align*} where $ x \in \mathbb{R}^N$, $u\in D^{s,p}(\mathbb{R}^N)$, $0<s<1$, $1<p<+\infty$, $N>sp$, $0<α<sp$, $0<β<sp$, $β\neqα$, $μ< μ_H := \inf_{u \in D^{s,p}(\mathbb{R}^N) …
We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). …
In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.
