高级非线性研究旨在发表关于非线性问题的论文，特别是那些涉及微分方程、动力系统和相关领域的论文。它还将发布这些领域在工程和科学问题上的新颖有趣的应用。提交给本刊的论文必须包含原创、及时和重要的成果。文章通常（但并非总是）按照收到最终副本时的顺序发表。

The journal "Advanced Nonlinear Studies" aims to publish research papers on nonlinear problems, particularly those involving differential equations, dynamical systems, and related fields. It will also feature novel and interesting applications of these areas in engineering and scientific issues. Submitted papers must contain original, timely, and significant findings. Articles are typically published in the order in which the final versions are received, although this is not always the case.

《Advanced Nonlinear Studies》期刊致力于发表关于非线性问题的研究论文，特别是涉及微分方程、动力系统及相关领域的研究。期刊还将发布这些领域在工程和科学问题上的新颖和有趣的应用。提交给本刊的论文必须包含原创性、时效性和重要性的研究成果。文章通常（但并非总是）按照收到最终稿件的顺序发表。

Advanced Nonlinear Studies

Abstract In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.

Wenxiong Chen

Yeyao Hu

Lingwei Ma

Moving planes and sliding methods for fractional elliptic and parabolic equations

移动平面法与滑动方法在分数阶椭圆和抛物方程中的应用

Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {L}_{p} chord measures is called the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {L}_{p} chord Minkowski problem in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {L}_{p} Brunn-Minkowski theory, which includes the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {L}_{p} Minkowski problem as a special case. This article solves the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {L}_{p} chord Minkowski problem when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> p\gt 1 and the symmetric case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:math> 0\lt p\lt 1 .

Dongmeng Xi

Deane Yang

Gaoyong Zhang

Yiming Zhao

The L p chord Minkowski problem

L p 和弦 Minkowski 问题

Abstract We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.

Mohammad Ghomi

Comparison formulas for total mean curvatures of Riemannian hypersurfaces

黎曼超曲面总平均曲率的比较公式

Abstract We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.

Jacob Bernstein

Rigidity properties of Colding–Minicozzi entropies

Colding-Minicozzi熵的刚性性质

Abstract We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

Pengfei Guan

Jiuzhou Huang

Jiawei Liu

Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces

非均匀完全非线性收缩的凸超曲面流

Abstract The purpose of this paper is to investigate the principal spectral theory and asymptotic behavior of the spectral bound for cooperative nonlocal dispersal systems, specifically focusing on the case where partial diffusion coefficients are zero, referred to as the partially degenerate case. We propose two sufficient conditions that ensure the existence of the principal eigenvalue in these partially degenerate systems. Additionally, we study the asymptotic behavior of the spectral bound for nonlocal dispersal operators with small and large diffusion coefficients, considering both non-degenerate and partially degenerate cases. Notably, we find a threshold-type result as the diffusion coefficients tend towards infinity in the partially degenerate case. Finally, we apply these findings to discuss the asymptotic behavior of the basic reproduction ratio in a viral diffusion model.

Lei Zhang

Principal spectral theory and asymptotic behavior of the spectral bound for partially degenerate nonlocal dispersal systems

部分退化非局部扩散系统的主谱理论与谱界的渐近行为

Abstract In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>ϕ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>H</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>ϕ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> <m:mtd columnalign="left"> <m:mtext>in</m:mtext> <m:mspace width="0.3333em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill &amp; \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill &amp; \text{in} {\mathbb{R}}^{3},\hfill \\ u{ &gt;}0,\quad \hfill &amp; \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ &gt; 0 is a small parameter, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mn>3</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $s\in \left(\frac{3}{4},1\right)$ , β &gt; 0, H is the Heaviside function, (−Δ) s u is the fractional Laplacian operator, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>V</m:mi> <m:mspace width="0.1em"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mspace width="0.1em"/> <m:mo>:</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.

Changyang Mu

Zhipeng Yang

Wei Zhang

Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity

分数阶薛定谔-泊松系统解的存在性与集中性研究：不连续非线性情形

Abstract In the present paper, we consider the coupled Schrödinger systems with critical exponent: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mi>λ</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∑</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mfenced close="|" open="|"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mtext> </m:mtext> <m:mtext> in </m:mtext> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1,2</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \\ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right),\quad i=1,2,\dots ,d,\quad \hfill \end{cases}$$ where d ≥ 2, β ii &gt; 0 for every i , β ij = β ji when i ≠ j , λ &gt; 0 is a parameter and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mtext>loc </m:mtext> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msubsup> <m:mfenced close=")" open="("> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:math> $0\le {V}_{i}\in {L}_{\text{loc\,}}^{\infty }\left({\mathbb{R}}^{N}\right)$ have a common bottom int <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${V}_{i}^{-1}\left(0\right)$ composed of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfenced> </m:math> ${\ell }_{0}\left({\ell }_{0}\ge 1\right)$ connected components <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mfenced close="}" open="{"> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> </m:math> ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{{\ell }_{0}}$ , where int <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${V}_{i}^{-1}\left(0\right)$ is the interior of the zero set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mfenced close="}" open="{"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">∣</m:mo> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mfenced> </m:math> ${V}_{i}^{-1}\left(0\right)=\left\{x\in {\mathbb{R}}^{N}\mid {V}_{i}\left(x\right)=0\right\}$ of V i . We study the existence of least energy positive solutions to this system which are trapped near the zero sets int <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${V}_{i}^{-1}\left(0\right)$ for λ &gt; 0 large for weakly cooperative case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mspace class="nbsp" width="0.3333em"/> <m:mi mathvariant="normal">s</m:mi> <m:mi mathvariant="normal">m</m:mi> <m:mi mathvariant="normal">a</m:mi> <m:mi mathvariant="normal">l</m:mi> <m:mi mathvariant="normal">l</m:mi> </m:mrow> </m:mfenced> </m:math> $\left({\beta }_{ij}{ &gt;}0 \mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\right)$ and for purely competitive case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mfenced> </m:math> $\left({\beta }_{ij}\le 0\right)$ . Besides, when d = 2, we construct a one-bump fully nontrivial solution which is localised at one prescribed components <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mfenced close="}" open="{"> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> </m:msubsup> </m:math> ${\left\{{{\Omega}}_{k}\right\}}_{k=1}^{\ell }$ for large λ .

Zongyan Lv

Zhongwei Tang

Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

陡势阱和临界指数耦合薛定谔系统的解

Abstract The Willmore flow is the negative gradient flow of the Willmore energy. In this paper, we consider a special kind of solutions to Willmore flow, which we call solitons, and investigate their geometric properties.

Pak Tung Ho

Juncheol Pyo

Solitons to the Willmore flow

Willmore流的孤子

Abstract In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ ( ρ ) = ρ θ , λ ( ρ ) = ( θ − 1) ρ θ in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ. , vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ &gt; 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>θ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> $\theta =\frac{2}{3}$ , N = 3 and γ &gt; 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>γ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>5</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> $\gamma =\frac{5}{3}$ , respectively.

Abbreviation	Adv. Nonlinear Stud.
Journal Impact	1.98
Quartiles(Global)	MATHEMATICS, APPLIED(Q1)

ISSN	1536-1365, 2169-0375
h-index	41

Publication Information	Publisher: Walter de Gruyter GmbH，Publishing cycle: Quarterly，Journal Type: journal，Open Access Journals: Yes
Basic data	Year of publication: 2001，Proportion of original research papers: 100.00%，Self Citation Rate:14.30%， Gold OA Rate: 78.13%
Journal Fees	$Details
Average review cycle	网友分享经验：>12周，或约稿
Average recruitment ratio	网友分享经验：容易

Advanced Nonlinear Studies

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