Ground state solutions for a modified fractional Schrödinger equation with critical exponent

In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations $${(-\mathrm{\Delta })}^{\alpha }u+\mu u+\kappa [{(-\mathrm{\Delta })}^{\alpha }{u}^2]u=\sigma |u{|}^{p-1}u+|u{|}^{q-2}u,x\in R^N,$$ where $N\ge 2$, $\alpha \in (0,1)$, $\mu$, $\sigma$ and $\kappa$ are positive parameters, $2<p+1<q\le 2_{\alpha }^{\ast }:=\frac{2N}{N-2\alpha }$, ${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian of order $\alpha$. For the case $2<p+1<q<2_{\alpha }^{\ast }$ and the case $q=2_{\alpha }^{\ast }$, the existence results of ground state solutions are given, respectively.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space ${\mathbb{R}}^n$ for $n\ge 3$. It is clarified that if w is small in ${\dot{B}}_{p_{*},q^{\prime }}^{-1+\frac{n}{p_{*}}}$ for and , then for every small initial disturbance $a\in {\dot{B}}_{p_0,q}^{-1+\frac{n}{p_0}}$ with and ($1/q+1/q^{\prime }=1$), there exists a unique solution $v(t)$ of the nonstationary Navier–Stokes equations on (0, ∞) with $v(0)=w+a$ such that ${\parallel v(t)-w\parallel }_{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})})$ and ${\parallel v(t)-w\parallel }_{{\dot{B}}_{p,q}^s}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{p})-\frac{s}{2}})$ as $t\to \infty$, for , , and small $s>0$.     

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

In this paper, we mainly study the local energy equation of the weak solutions of the compressible isentropic MHD equation defined on ${\mathrm{??}}^3$. We prove that the regularity of the solution is sufficient to guarantee the balance of the total energy in the $B_3^{\alpha ,\infty }((0,T)\times {\mathrm{??}}^3)$ space. We adopt a variant of the method of Feireisl et al.     

A multiperiod model for estimating the cost of deposit insurance under the Basel III minimum capital requirement

Under the Basel III regime, a commercial bank is considered adequately capitalized if it maintains a ratio of capital to total risk-weighted assets or capital adequacy ratio (CAR) of at least 8%. We model a commercial bank that complies with Basel III's minimum capital requirement on an interval $[0,T]$ for $T>0$. The bank model is achieved via a specific rate of capital influx that fixes the bank's CAR at the minimum prescribed level of 8%. On the basis of this capital influx rate, we derive models for the bank's asset portfolio and capital dynamics required for maintaining the CAR at the minimum prescribed level. For the aforementioned bank, we further study a deposit insurance (DI) pricing problem with a coverage horizon equal to $T$ years. More specifically, we employ a multiperiod DI pricing model to approximate the cost of DI for the bank on the interval $[0,T]$, where the constant (minimum) CAR is maintained. We study the behaviours of the models leading to the constant (minimum) CAR, and the behaviour of the DI premium estimate by means of numerical simulations. In the simulation study pertaining to the DI premium estimate specifically, we determine the effects of changes in the bank's initial leverage level (deposit-to-asset ratio), the DI coverage horizon, and the volatility of the asset portfolio on the DI premium estimate.     

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

Multibump solutions for nonlinear Schrödinger-Poisson systems

In this paper, we study the following Schrödinger-Poisson equations: $$\left\{\begin{array}{ll}-{\epsilon }^2\mathrm{\Delta }u+V(x)u+K(x)\varphi u=|u{|}^{p-2}u,& x\in {\mathbb{R}}^3,\\ -{\epsilon }^2\mathrm{\Delta }\varphi =K(x)u^2,& x\in {\mathbb{R}}^3,\end{array}\right.$$ where $p\in (4,6)$, $\epsilon >0$ is a parameter and $V$ and $K$ satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.     

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether $\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}$ takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range $0?\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}?1$. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.     

On the fixed part of pluricanonical systems for surfaces

We show that $|m{K}_X|$ defines a birational map and has no fixed part for some bounded positive integer m for any $\frac{1}{2}$-lc surface X such that $K_X$ is big and nef. For every positive integer $n\ge 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, $\mathrm{mld}(X_{n,i})>\frac{1}{n}$ for every i, ${\lim }_{i\to +\infty }\mathrm{mld}(X_{n,i})=\frac{1}{n}$, and for any positive integer m, there exists i such that $|m{K}_{X_{n,i}}|$ has nonzero fixed part. These results answer the surface case of a question of Xu.     

Degrees of categoricity of trees and the isomorphism problem

In this paper, we show that for any computable ordinal α, there exists a computable tree of rank $\alpha +1$ with strong degree of categoricity ${\mathbf{0}}^{(2\alpha )}$ if α is finite, and with strong degree of categoricity ${\mathbf{0}}^{(2\alpha +1)}$ if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with strong degree of categoricity ${\mathbf{0}}^{(\alpha )}$ (which equals ${\mathbf{0}}^{(2\alpha )}$). It follows from our proofs that, for every computable ordinal $\alpha >0$, the isomorphism problem for trees of rank α is ${\mathrm{\Pi }}_{2\alpha }^0$‐complete.     

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions

We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions $B_{p,\theta }^{\mathit{r}}({\mathbb{T}}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions $QC({\mathbb{T}}^d)$. We also showed that for $2\le p\le \infty$, , $r_1>\frac{1}{2}$, $d\ge 2$, the estimate of the corresponding asymptotic characteristic is exact in order.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

We prove $L^p$ estimates, $1\le p\le \infty$, for solutions to the tangential Cauchy–Riemann equations ${\overline{?}}_{b}u=?$ on a class of infinite type domains $\mathrm{\Omega }?{\mathbb{C}}^2$. The domains under consideration are a class of convex ellipsoids, and we show that if $?$ is a ${\overline{?}}_b$‐closed (0,1)‐form with coefficients in $L^p$, then there exists an explicit solution u satisfying ${\parallel u\parallel }_{L^p(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^p(b\mathrm{\Omega })}$. Moreover, when $p=\infty$, we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the $\overline{?}$‐equation, that is, given a smooth (0,1)‐form ? on $b\mathrm{\Omega }$ with an L¹‐boundary value, we can solve the Cauchy–Riemann equation $\overline{?}u=?$ so that ${\parallel u\parallel }_{L^1(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^1(b\mathrm{\Omega })}$ where C is independent of $u$ and ?. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.     

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters $t_1,\dots ,t_k$. A formula in this language defines a parametric set $S_{\mathbf{t}}?{\mathbb{Z}}^d$ as $\mathbf{t}$ varies in ${\mathbb{Z}}^k$, and we examine the counting function $|S_{\mathbf{t}}|$ as a function of t . For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming $\mathbf{P}\ne \mathbf{NP}$) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1,t_2}|$ is not even polynomial‐time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_{\mathbf{t}}?{\mathbb{Z}}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_{\mathbf{t}}|$ is always polynomial‐time computable in the size of t , and in fact can be represented using the gcd and similar functions.     

Decidable fan theorem and uniform continuity theorem with continuous moduli

The uniform continuity theorem $(\mathsf{UCT})$ states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive mathematics, $\mathsf{UCT}$ is strictly stronger than the decidable fan theorem $(\mathsf{DFT})$, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real-valued functions as type-one functions. However, the precise relation between such type-one functions and continuous real-valued functions (usually described as type-two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real-valued function on [0, 1], and show that real-valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that $\mathsf{DFT}$ is equivalent to the statement that every pointwise continuous real-valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that $\mathsf{DFT}$ is equivalent to a similar principle for real-valued functions on the Cantor space ${\{0,1\}}^{\mathbb{N}}$. These results extend Berger's [2] characterisation of $\mathsf{DFT}$ for integer-valued functions on ${\{0,1\}}^{\mathbb{N}}$ and unify some characterisations of $\mathsf{DFT}$ in terms of functions having continuous moduli.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.