Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

In this paper, we study the existence of minimizers for $$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$ on the constraint $$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$ , where c >  0 is a given parameter. In the range ${p \in [3,\frac{10}{3}]}$ , we explicit a threshold value of c >  0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c >  0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

On the approximation of conjugate functions from Holder classes by Fejér means

Получены асимптотич еские равенства для в еличин гдеr≧0 — целое, ω(t) — выпу клый модуль непрерыв ности и $$\bar \sigma _n (f;x) = - \frac{1}{\pi } \mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2}ctg\frac{t}{2} - \frac{1}{{4(n + 1)}}\frac{{\sin (n + 1)t}}{{\sin ^2 \tfrac{1}{2}t}}} \right)dt$$ сумма Фейера функцииf(х), сопряженной сf(x).     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

On the Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space

Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space $H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with n∈? and 1<q≤p<∞, which coincides with the usual critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$ in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant C _p,q depending only on p and q such that GN $$ \|f\|_{{\mathcal{M}}_{r,\frac{q}{p}r}} \leq C_{p,q}r\|f \|_{{\mathcal{M}}_{p,q}}^{\frac{p}{r}}\bigl\|(-\Delta)^{\frac{n}{2p}} f\bigr\|_{{\mathcal{M}}_{p,q}}^{1-\frac{p}{r}} $$ for all $u\in H\mathcal{M}^{\frac{n}{p}}_{p,q}( \mathbb{R}^{n})$ and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space $H^{\frac{n}{p}}_{p}(\mathbb{R}^{n})$ , the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765–769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531–540, 1991) and Ozawa (J. Func. Anal. 127:259–269, 1995) with the growth order $r^{1-\frac{1}{p}}$ as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

An improvement on Waring?CGoldbach problem for unlike powers

Let p _i be prime numbers. In this paper, it is proved that for any integer k?R5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$ . This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1?C4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ . This method can also be used for some other similar forms.     

A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Asymptotics of the approximation of continuous and differentiable functions by the singular integrals of de la Vallée Poussin

The complete asymptotic developments in powers of 1/n are derived for quantities characterizing approximation by singular integrals of de la Vallée Poussin $$V_n (f:x) = \frac{1}{{\Delta _n }}\int_{ - \pi }^\pi {f(x + t)} \cos ^{2n} \frac{t}{2}dt;\Delta _n = \int_{ - \pi }^\pi {\cos ^{2n} \frac{t}{2}dt}$$ of the function classes Lipa, 0w ^(r), r?1 an integer.     

Singularity analysis of Ginzburg–Landau energy related to p-wave superconductivity

The following Ginzburg–Landau energy in the absence of a magnetic field $$E_\varepsilon(\psi) = \int\limits_G\left[\frac{1}{2}|\nabla\psi|^2 + \frac{1}{4\varepsilon^2}(1-|\psi|^2)^2\right]{\rm d}x$$ was well studied during recent twenty years. Here, ${G \subset \mathbf{R}^2}$ is a bounded smooth domain, ${\psi}$ is an order parameter, ${\varepsilon >0 }$ . In particular, several global properties including the weighted energy estimation, the concentration compactness properties and the quantization effect of the energy had been established. This paper is concerned with another Ginzburg–Landau type free energy associated with p-wave superconductivity $$E_\varepsilon (\psi, u; G) = \frac{1}{2} \int\limits_G(|\nabla \psi|^2 + |\nabla u|^2 - |\nabla|\psi||^2){\rm d}x + \frac{1}{4\varepsilon^2} \int\limits_G(1-|\psi|^2)^2{\rm d}x.$$ Here, u is also an order parameter. We will prove that those global properties still hold for this more complicated energy functional. Such global properties describe the locations of the regular and the singular domains, and also show the convergence relation between the Ginzburg–Landau minimizers and the harmonic maps.     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

A logarithmically improved regularity criterion for the Navier–Stokes equations

In this note we prove a logarithmically improved regularity criterion in terms of the Besov space norm for the Navier–Stokes equations. The result shows that if a mild solution u satisfies ${\int_{0}^{T}\frac{\|u (t,\cdot)\|_{{\dot{B}}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}}{1+\ln(e+\| u(t,\cdot)\|_{H^{s}})}\text{d}t < \infty}$ for some 0?≤ r?<?1 and ${s\geq\frac{n}{2}-1}$ , then u is regular at t?=?T.     

Riesz Means on Graphs and Discrete Groups

Suppose that Γ is a weighted graph or a discrete group. Let $m_{\alpha,R}(\lambda )=\big(1-\big|\frac{\lambda}{R}\big|\big)_{+}^{\alpha}$ be the Riesz means and let Δ be the discrete Laplacian on Γ. We prove that if D is the homogeneous dimension of Γ then the operator m _α,R (Δ) is bounded on L ^p , provided that $\alpha>D|\frac{1}{p}-\frac{1}{2}|$ .     

Boundedness of Singular Integrals in Journé’s Class on Weighted Multiparameter Hardy Spaces

In this paper, we obtain the boundedness of singular integral operators T in Journé’s class on weighted multiparameter Hardy spaces \(H^{p}_{w}\) of arbitrary k number of parameters (k≥3) under the assumption that \(T^{\ast}_{i}(1)=0\) , i=1,…,k, and the kernel of T has a regularity of order ?>0, where \(w \in A_{r}(\Bbb{R}^{n_{1}}\times \cdots \times \Bbb{R}^{n_{k}}), r \geq 1\) and \(\max\{ \frac{r n_{1} }{n_{1}+\varepsilon}, \ldots, \frac{r n_{k} }{n_{k}+\varepsilon}\} .     

Nonexistence of Positive Solutions for an Integral Equation Related to the Hardy-Sobolev Inequality

Let α and s be real numbers satisfying 0<s<α<n. We are concerned with the integral equation $$u(x)=\int_{R^n}\frac{u^p(y)}{|x-y|^{n-\alpha}|y|^s}dy, $$ where \(\frac{n-s}{n-\alpha}< p< \alpha^{*}(s)-1\) with \(\alpha^{*}(s)=\frac{2(n-s)}{n-\alpha}\) . We prove the nonexistence of positive solutions for the equation and establish the equivalence between the above integral equation and the following partial differential equation $$\begin{aligned} (-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{-s}u^p. \end{aligned}$$      

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Nonexistence of quasi-harmonic spheres with large energy

The absence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let (N, h) be a complete noncompact Riemannian manifold. Assume the universal covering of (N, h) admits a nonnegative strictly convex function with polynomial growth. Then there is no non-constant quasi-harmonic sphere ${u:\mathbb{R}^n\rightarrow N}$ such that $$\lim_{r \rightarrow \infty}r^ne^{-\frac{r^2}{4}}\int \limits_{|x|\leq r}e^{-\frac{|x|^2}{4}}|\nabla u|^2{\text {d}}x\,=\,0.$$ This generalizes a result of the first author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus comparatively elementary.     

Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems: $$(P)\quad \quad \left\{\begin{array}{ll}{-\Delta_{ap}u + V(x)|x|^{-ap^*} |u|^{p-2} u=K(x)f(x, u), {\rm in} \, R^N,}\\ {u > 0, {\rm in} \, R^N , \, u \in \mathcal{D}^{1,p}_a}{(R^N)},\end{array}\right. $$ denotes the Hardy-Sobolev’s \({{-\Delta_{ap}u = - div(|x|^{-ap}|\nabla u|^{p-2} \nabla u), 1 < p < N, -\infty < a < \frac{N-p}{p}, a \leq e \leq a+1, d=1+a-e}}\) , and \({{p^* := p^*(a,e)=\frac{Np}{N-dp}}}\) denotes the Hardy-Sobolev’s critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.