Finite computable dimension and degrees of categoricity

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below ${\mathbf{0}}^{″}$, and not above ${\mathbf{0}}^{\prime }$. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form $[{\mathbf{0}}^{(\alpha )},{\mathbf{0}}^{(\alpha +1)}]$ for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if ${\mathbf{d}}_0$, ${\mathbf{d}}_1$, and ${\mathbf{d}}_2$ are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each ${\mathbf{d}}_i<{\mathbf{d}}_0\oplus {\mathbf{d}}_1\oplus {\mathbf{d}}_2$. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$\{\begin{array}{cc}\hfill & ?{\epsilon }^2\mathrm{\Delta }u+V(x)u+{\lambda }_1|u{|}^{2}u+{\lambda }_2(K?|u{|}^2)u=0\text{ in }{\mathbb{R}}^3,\hfill \\ \hfill & u>0,u\in H^1({\mathbb{R}}^3),\hfill \end{array}$

where ε is a small positive parameter, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ? denotes the convolution, $K(x)=\frac{1?3{\cos }^2?\theta }{|x{|}^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under certain assumptions on $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}^2$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local minima of V as $\epsilon \to 0$. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.     

Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

Concentration on curves for a nonlinear Schrödinger problem with electromagnetic potential

We prove the existence of solutions to the nonlinear Schrödinger equation ${\epsilon }^2{(i\mathrm{?}+\mathit{A})}^{2}u+V(y)u?|u{|}^{p?1}u=0$ in ${\mathbb{R}}^2$ with a magnetic potential $\mathit{A}=(A_1,A_2)$. Here V represents the electric potential, the index p is greater than 1. Along some sequence $\{{\epsilon }_n\}$ tending to zero we exhibit complex-value solutions that concentrate along some closed curves.     

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, $u_t+u_{xxxx}+{?}_{xx}{u}_x^2=0$, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition $u_x\in L^{q^{\prime }}{L}^q(Q)$ is satisfied, where $q,q^{\prime }\in [1,\infty ]$ are such that either $1/q+4/q^{\prime }<1$ or $1/q+4/q^{\prime }=1$, $q^{\prime }<\infty$.     

Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term$\{\begin{array}{cc}?\mathrm{\Delta }u+u={\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u}),u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ \frac{?u}{?\nu }=0,\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^2$ is a bounded, smooth domain, $\epsilon >0$, ${\alpha }_1,...,{\alpha }_n\in (0,\infty )?\mathbb{N}$, and $q_1,...,q_n\in \mathrm{\Omega }$ are fixed. Given any two non-negative integers $k,l$ with $k+l?1$, it is shown that, for sufficiently small $\epsilon >0$, there exists a solution $u_{\epsilon }$ for which ${\epsilon }^2|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}({\text{e}}^u?{\text{e}}^{?u})$ asymptotically (i.e. the limit as $\epsilon \to 0$) develops $k+n$ interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function $k(x)=|x?q_1{|}^{2{\alpha }_1}?|x?q_n{|}^{2{\alpha }_n}$.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

The Orlicz inequality for multilinear forms

The Orlicz $({?}_2,{?}_1)$-mixed inequality states that${\left(\sum _{j_1=1}^n{\left(\sum _{j_2=1}^n|A(e_{j_1},e_{j_2})|\right)}^2\right)}^{\frac{1}{2}}\le \sqrt{2}6A6$ for all bilinear forms $A:{\mathbb{K}}^n\times {\mathbb{K}}^n\to \mathbb{K}$ and all positive integers n, where ${\mathbb{K}}^n$ denotes ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with ${\mathbb{K}}^n$ endowed with ${?}_p$ norms for all $p\in [1,\infty ]$.     

On the boomerang uniformity of a class of permutation quadrinomials over finite fields

Let ${\mathbb{F}}_{2^n}$ be a finite field with $2^n$ elements and $f_{\underset{\_}{c}}(x)=c_0{x}^{2^m(2^k+1)}+c_1{x}^{2^{m+k}+1}+c_2{x}^{2^m+2^k}+c_3{x}^{2^k+1}\in {\mathbb{F}}_{2^n}[x]$, where n, m and k are positive integers with $n=2m$ and $\gcd ?(m,k)=e$. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of $f_{\underset{\_}{c}}(x)$ by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result from the case of m being odd and $e=1$ to that of both $m/e$ and $k/e$ being odd.     

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

Classification of cyclic codes over a non-Galois chain ring Zp[u]/〈u3〉

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$ and four types of non-principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$, which are associated with cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$. We then obtain a mass formula for cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$.