We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral...

Let A = -Δ + V be a Schrödinger operator on ${ℝ}^d$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_A^p$ if the maximal function $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, where ${T_t}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_A^p$ admits a special atomic decomposition.

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.

Let $G$ be a graph of order $n$ and $\lambda \left(G\right)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$. One of the main results of the paper is the following theorem: Let $k\ge 2,$ $n\ge k^3+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta \left(G\right)\ge k.$ If $$\lambda \left(G\right)\ge n-k-1,$$ then $G$ has a Hamiltonian cycle, unless $G=K_1\vee (K_{n-k-1}+K_k)$ or $G=K_k\vee (K_{n-2k}+{\overline{K}}_k).$

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the...

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t\&gt;0$ the flow map at time $t$ is discontinuous as a map from ${𝒞}^{\infty }(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ The argument relies...

Let ${\left(F_n\right)}_{n\in \mathbb{N}}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of ${\left(F_n\right)}_{n\in \mathbb{N}}$, we will prove that if $g\in L^p\left({ℝ}^N\right)$, $1<p<+\infty$, satisfies ${lim\; inf}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy<+\infty$, then $g\in W^{1,p}\left({ℝ}^N\right)$ and moreover ${lim}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy=K_{N,p}{\int }_{{ℝ}^N}{\left|\nabla g\left(x\right)\right|}^{p}dx$, where $K_{N,p}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m,p}\left(\Omega \right)$ into $L^S\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m,p}\left(\Omega \right)$ into $L^{\phi }\left(\Omega \right)$...............................................................

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d}f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$ $(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge n/2$. The Hardy type space $H_{{ℒ}_2}^1$ is defined in terms of the maximal function with respect to the semigroup $\left\{{\mathrm{e}}^{-t{ℒ}_2}\right\}$ and it is identical to the Hardy space $H_{{ℒ}_1}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator ${ℛ}_b=bℛf-ℛ\left(bf\right)$ generated by the Riesz transform $ℛ={\nabla }^2{ℒ}_2^{-1/2}$, where $b\in {\mathrm{BMO}}_{\theta }\left(\rho \right)$, which is larger...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions...

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$). ...

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: $$\frac{\mathrm{d}}{\mathrm{d}t}u+\mathrm{i}\left(-\Delta +V\left(x\right)\right)u=\nu \left(\Delta u-{\gamma }_R{\left|u\right|}^{2p}u-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u\right)+\sqrt{\nu }\eta (t,x).(*)$$ The force $\eta$ is white in time and smooth in $x$; the potential $V\left(x\right)$ is typical. We are concerned with the limiting, as $\nu \to 0$, behaviour of solutions on long time-intervals $0\le t\le {\nu }^{-1}T$, and with behaviour of these solutions under the double limit $t\to \infty$ and $\nu \to 0$. We show that these two limiting behaviours may be described in terms of solutions for the($*$) which is a well...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta –1}V\left(B^{\beta }\right)$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...