Mathematische Nachrichten最新文献

{"title":"Good Real Perturbations of Simple Corank One Map Germs From \u0000 \u0000 \u0000 R\u0000 3\u0000 \u0000 $mathbb {R}^3$\u0000 to \u0000 \u0000 \u0000 R\u0000 3\u0000 \u0000 $mathbb {R}^3$","authors":"A. J. Miranda,&nbsp;M. J. Saia,&nbsp;T. O. Souza","doi":"10.1002/mana.70216","DOIUrl":"https://doi.org/10.1002/mana.70216","url":null,"abstract":"<p>The existence of good real perturbations of map germs is one of the most interesting open questions in singularity theory. In this paper, we investigate this question for all simple corank one map germs from <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math> to <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^3$</annotation>\u0000 </semantics></math>, we give a complete description of those that admit a good real perturbation. The main source of information to study these questions is obtained from the multiple point sets of these germs.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2805-2835"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70216","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149100686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Recovering the Shape of a Quantum Tree by Scattering Data","authors":"O. Boyko,&nbsp;Y. Latushkin,&nbsp;V. Pivovarchik","doi":"10.1002/mana.70228","DOIUrl":"https://doi.org/10.1002/mana.70228","url":null,"abstract":"<div>\u0000 \u0000 <p>We consider a scattering problem generated by the Sturm–Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math>-function of the scattering problem and the eigenvalues of the operators associated with the compact tree.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"3036-3065"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149100133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matrix Representations of Eigenparameter-Dependent Dirac Operators With Finite Spectrum","authors":"Jinming Cai,&nbsp;Sijie Wang,&nbsp;Kun Li","doi":"10.1002/mana.70215","DOIUrl":"https://doi.org/10.1002/mana.70215","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, the finite spectrum problem of Dirac operators with an eigenparameter contained in the boundary conditions is studied. First, the existence of finite spectrum under certain conditions is proved, that is, for each nonnegative integer <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>, we construct a class of regular Dirac operators, each of which has at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <mi>m</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n=2m+2$</annotation>\u0000 </semantics></math> eigenvalues. Second, we identify a class of Dirac equations such that every Dirac operator formed by such an equation together with eigenparameter-dependent boundary conditions can be reformulated as an equivalent finite-dimensional matrix eigenvalue problem. Third, for any matrix eigenvalue problem of a prescribed type paired with eigenparameter-dependent boundary conditions, we construct a family of Dirac operators equipped with the specified boundary conditions, each of which is equivalent to the given matrix eigenvalue problem. Here, equivalence signifies that the two problems share identical sets of eigenvalues.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2789-2804"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149100704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Controllability of a Parabolic–Elliptic System Governed by a Bilinear Control","authors":"J. Límaco,&nbsp;A. R. Lopes,&nbsp;L. Prouvée","doi":"10.1002/mana.70219","DOIUrl":"https://doi.org/10.1002/mana.70219","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper deals with the analysis of the internal bilinear control for two nonlinear parabolic–elliptic systems. In a first case, we study the exact controllability to the trajectories for the system in which the control function acts on the parabolic equation. In the second case, when the control function acts on the elliptic equation, we control to the trajectories the parabolic component while keeping the elliptic component uniformly bounded. Our approach, in both cases, relies on an application of Liusternik's inverse mapping theorem which demands the proof of suitable Carleman estimate.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2867-2893"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149099633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Almost Periodic Solutions of a Keller–Segel–Fluid System With Singular Data","authors":"Dam Thi Ngoc Van,&nbsp;Pham Truong Xuan","doi":"10.1002/mana.70222","DOIUrl":"https://doi.org/10.1002/mana.70222","url":null,"abstract":"<div>\u0000 \u0000 <p>We study the existence and exponential stability of almost periodic mild solutions for the Patlak–Keller–Segel–Navier–Stokes system on the real hyperbolic manifold <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathbb {H}^d(mathbb {R}), (dgeqslant 3)$</annotation>\u0000 </semantics></math> in the framework of weak-<span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msup>\u0000 <annotation>$L^p$</annotation>\u0000 </semantics></math> spaces. First, we establish the dispersive estimates and Yamazaki-type estimates for both scalar and vectorial heat semigroups on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathbb {H}^d(mathbb {R})$</annotation>\u0000 </semantics></math>. Leveraging by these interpolation estimates, we obtain the global existence of mild solutions for the corresponding linear systems on the whole line time-axis. Then, we prove a Massera-type principle which guarantees the existence of almost periodic mild solutions for the linear systems. Next, we combine the linear results with fixed-point arguments to obtain the existence of almost periodic mild solutions for the semilinear systems. Finally, we prove an exponential stability for such mild solutions.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2944-2965"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149099634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Roughness of Nonuniform (ρ−, ρ+)-Exponential Dichotomy","authors":"Yeming Chen,&nbsp;Weinian Zhang,&nbsp;Linfeng Zhou","doi":"10.1002/mana.70218","DOIUrl":"https://doi.org/10.1002/mana.70218","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this paper, we introduce the concept of nonuniform &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(rho _-, rho _+)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-exponential dichotomies to formulate an asymptotic behavior weaker than the known nonuniform &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;annotation&gt;$rho$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-exponential dichotomy. We discuss the roughness of such dichotomies on the whole line and the half line separately without global invertibility of the evolution family. Our results extend the known results to nonuniform &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;annotation&gt;$rho$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-exponential dichotomies in the case that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;lim&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mo&gt;±&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$lim _{trightarrow pm infty }rho _{-}(t)/rho _{+}(t)=0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; or &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;lim&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mo&gt;±&lt;/mo&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ρ&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2836-2866"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149099636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hyperbolic Generalized Framed Surfaces in Hyperbolic 3-Space","authors":"Donghe Pei,&nbsp;Masatomo Takahashi,&nbsp;Anjie Zhou","doi":"10.1002/mana.70221","DOIUrl":"https://doi.org/10.1002/mana.70221","url":null,"abstract":"<div>\u0000 \u0000 <p>Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations among hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2918-2943"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149099839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Test Function Method for Weakly Coupled Systems With Derivative-Type Nonlinearity","authors":"Marcello D'Abbicco,&nbsp;Antonio Lagioia","doi":"10.1002/mana.70227","DOIUrl":"https://doi.org/10.1002/mana.70227","url":null,"abstract":"<p>We consider a two by two system of inequalities which include fractional powers of the Laplace operator, coupled through a semilinear term of derivative type. We prove the nonexistence of global-in-time weak solutions for powers below the critical curve and possibly on the critical curve.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"3015-3035"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70227","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149100134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Combinatorial Bounds for the Total Tjurina Numbers of Certain Curves and Surfaces With Isolated Singularities","authors":"Piotr Pokora","doi":"10.1002/mana.70217","DOIUrl":"https://doi.org/10.1002/mana.70217","url":null,"abstract":"<div>\u0000 \u0000 <p>We investigate combinatorial bounds for the total Tjurina numbers of some plane curve arrangements. Focusing on arrangements of lines and conics in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {P}^2$</annotation>\u0000 </semantics></math> that admit only ordinary quasi-homogeneous singularities, we derive new structural inequalities governing the distribution of multiple intersection points. As a consequence, we establish sharp lower bounds for the total Tjurina numbers of free line arrangements with bounded maximal multiplicity and, more generally, for free conic-line arrangements. In particular, we show that for a free arrangement of <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> lines and <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> conics, the total Tjurina number grows at least quadratically in <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>, and we demonstrate that this bound is sharp. As an application of these planar results, we construct a special family of surfaces in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>$mathbb {P}^{3}$</annotation>\u0000 </semantics></math> with only isolated singularities and arbitrarily large total Tjurina numbers. This provides new lower bounds for the total Tjurina numbers of certain hypersurfaces that are independent of detailed homological data.</p></div>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2751-2759"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149100696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ground-State Solutions for a Singular Elliptic Kirchhoff–Boussinesq Equation in \u0000 \u0000 \u0000 R\u0000 4\u0000 \u0000 $mathbb {R}^4$","authors":"Yony R. S. Leuyacc,&nbsp;Romulo D. Carlos,&nbsp;Segundo M. A. Salirrosas","doi":"10.1002/mana.70225","DOIUrl":"https://doi.org/10.1002/mana.70225","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This work investigates the existence of nontrivial ground-state solutions for singular elliptic Kirchhoff–Boussinesq equations:\u0000 &lt;/p&gt;&lt;div&gt;&lt;span&gt;&lt;span&gt;&lt;!--FIGURE--&gt;&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;±&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mtext&gt;in&lt;/mtext&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$begin{equation*} Delta ^{2} u pm Delta _{p} u + V(x)u= frac{f(x,u)}{|x|^{beta }} mbox{in} mathbb {R}^{4}, end{equation*}$$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/div&gt;where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$2&lt; p&lt; 4$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 10","pages":"2966-2982"},"PeriodicalIF":1.0,"publicationDate":"2026-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149099750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}