Journal of Mathematical Analysis and Applications最新文献

{"title":"Existence and multiplicity of solutions to quasilinear Dirac-Poisson systems with varying potentials","authors":"Minbo Yang ,&nbsp;Fan Zhou","doi":"10.1016/j.jmaa.2025.130374","DOIUrl":"10.1016/j.jmaa.2025.130374","url":null,"abstract":"<div><div>In this paper, we consider the following Dirac-Poisson system:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>i</mi><mi>ε</mi><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msub><mi>w</mi><mo>−</mo><mi>a</mi><mi>β</mi><mi>w</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>w</mi><mo>−</mo><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>ψ</mi><mi>w</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>|</mo><mi>w</mi><mo>|</mo><mo>)</mo><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>Δ</mi><mi>ψ</mi><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>5</mn></mrow></msup><msub><mrow><mi>Δ</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>ψ</mi><mo>=</mo><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>x</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>∂</mo><mo>/</mo><mo>∂</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, <span><math><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>, <span><math><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span> is a constant and the operator <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is the 4-Laplacian operator, defined as <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>ψ</mi><mo>:</mo><mo>=</mo><mi>d</mi><mi>i</mi><mi>v</mi><mo>(</mo><mo>|</mo><mi>∇</mi><mi>ψ</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>∇</mi><mi>ψ</mi><mo>)</mo></math></span>. For the subcritical and the critical cases, we prove the existence and multiplicity of solutions and get that the solutions of the quasilinear Dirac-Poisson system converge to solutions of a semilinear Dirac-Poisson system as <em>ε</em> tending to 0. Some further properties of the solutions are also obtained.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130374"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928542","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":"Bicomplex Hardy classes of solutions to higher-order Vekua equations","authors":"William L. Blair","doi":"10.1016/j.jmaa.2026.130397","DOIUrl":"10.1016/j.jmaa.2026.130397","url":null,"abstract":"<div><div>We extend representation formulas for functions in Hardy classes of solutions to higher-order iterated Vekua equations to Hardy classes of bicomplex-valued functions that solve a bicomplex version of the Vekua equation or its higher-order generalizations. Using these representations, we show that functions in these bicomplex-valued Hardy classes have nontangential boundary values and boundary values in the sense of distributions.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130397"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978788","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":"Full stability characterization of shearing Cattaneo-Timoshenko systems","authors":"Eduardo H. Gomes Tavares ,&nbsp;Anderson J.A. Ramos ,&nbsp;Marcio A. Jorge Silva ,&nbsp;JinYun Yuan","doi":"10.1016/j.jmaa.2026.130452","DOIUrl":"10.1016/j.jmaa.2026.130452","url":null,"abstract":"<div><div>We provide a complete analytical and numerical characterization of the stability behavior for a shearing Cattaneo-Timoshenko system governed by a recently proposed stability number, denoted herein by <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span>. Unlike previous results restricted to energy-based approaches or specific parameter configurations, we prove that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> is not only sufficient but also necessary for exponential stability. Otherwise, the system exhibits optimal algebraic decay rate <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> for regular initial data. The theoretical results are aligned with a proper numerical analysis where a finite difference scheme confirms the characterization of stability in terms of <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> and its effectiveness in predicting the complete system's asymptotic behavior.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130452"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078314","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":"Volume preserving Willmore flow in a generalized Cahn-Hilliard flow","authors":"Yuan Chen","doi":"10.1016/j.jmaa.2026.130451","DOIUrl":"10.1016/j.jmaa.2026.130451","url":null,"abstract":"<div><div>We investigate the mass-preserving <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-gradient flow associated with a generalized Cahn–Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is small. For well-prepared initial data, we rigorously prove that, as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>, solutions of the diffuse-interface model converge to the <em>volume-preserving Willmore flow</em> in arbitrary spatial dimensions <span><math><mi>N</mi><mo>≥</mo><mn>2</mn></math></span>. The proof incorporates matched asymptotic expansions, spectrum and energy estimates to establish the convergence of the order parameter away from the interface, alongside a precise motion law derivation for the limiting interface. This result extends the analysis of Fei and Liu <span><span>[16]</span></span> from two- and three-dimensional settings to general <em>N</em>-dimensional domains, and it applies to a broad class of symmetric double-well potentials beyond the classical quartic form.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130451"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023451","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 instability and stability of a quasi-linear hyperbolic-parabolic model for vasculogenesis","authors":"Qing Chen ,&nbsp;Huaqiao Wang ,&nbsp;Guochun Wu","doi":"10.1016/j.jmaa.2025.130375","DOIUrl":"10.1016/j.jmaa.2025.130375","url":null,"abstract":"<div><div>In this paper, we are concerned with the instability and stability of a quasi-linear hyperbolic-parabolic system modeling vascular networks. Under the assumption that the critical cell density satisfies <span><math><mfrac><mrow><mi>ν</mi><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></mrow><mrow><mi>γ</mi><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></mfrac><mo>&lt;</mo><mi>β</mi></math></span>, we show that the steady-state is linear unstable (i.e., the linear solution grows in time in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>) by constructing an unstable solution. Then based on the lower growth estimates on the solution to the linear system, we prove that the steady-state is nonlinear unstable in the sense of Hadamard. On the contrary, if <span><math><mfrac><mrow><mi>ν</mi><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></mrow><mrow><mi>γ</mi><mover><mrow><mi>ρ</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></mfrac><mo>&gt;</mo><mi>β</mi></math></span>, we establish the global existence for small perturbations and the optimal convergent rates for all-order derivatives of the solution by slightly getting rid of the condition proposed in Liu et al. (2022) <span><span>[16]</span></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130375"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928547","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":"Bessel functions and quantum harmonic oscillator","authors":"Irina So","doi":"10.1016/j.jmaa.2025.130376","DOIUrl":"10.1016/j.jmaa.2025.130376","url":null,"abstract":"<div><div>New relations for Bessel functions are obtained by applying the “creation” and “annihilation” operators to non-standard solutions of the Schrödinger equation of the harmonic oscillator. Involving Bessel functions, the solutions are either non-normalizable, or normalizable but non-smooth at <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span>. None of them is an eigenfunction of the “creation” or “annihilation” operator with zero eigenvalue, so negative energy eigenvalues arise. In this connection, the work explains how non-normalizability and non-smoothness of the solutions can cause occurrence of negative energy eigenvalues. As to physical application of the solutions, we consider a harmonic oscillator with a step at <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span>, in which two halves are shifted relative to each other. Its wave function is composed of normalizable Bessel-type solutions for which the possibility of being smoothly sewed at <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span> is shown via the obtained relations. The solutions which initially corresponded to negative eigenvalues now describe tunnelling through the potential step.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130376"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928544","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":"Existence of solutions to diffusive logistic models with transmission conditions on normal derivatives","authors":"Zhiyuan Wen,&nbsp;Lijuan Zhou","doi":"10.1016/j.jmaa.2025.130388","DOIUrl":"10.1016/j.jmaa.2025.130388","url":null,"abstract":"<div><div>In this paper, we will consider a boundary value problem of the diffusive logistic equation which is defined on the <em>N</em>-dimensional unit ball. The boundary conditions include not only the Neumann condition on the outer sphere but also an interior jump condition involving the normal derivatives on a sphere located within the ball. To establish the existence and uniqueness of the solution, we first analyze an approximate problem associated with the jump condition. We will show that this approximate problem admits a unique solution, which can be expressed as a power series in terms of the parameter in the jump condition. Secondly, we will show this power series converges to the unique solution of the original jump problem.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130388"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928541","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":"Local Hölder regularity for quasilinear elliptic equations with mixed local-nonlocal operators, variable exponents, and weights","authors":"Juan Pablo Alcon Apaza","doi":"10.1016/j.jmaa.2026.130417","DOIUrl":"10.1016/j.jmaa.2026.130417","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;div&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt; in &lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, is a bounded domain. The nonlocal operator is defined by&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;.&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;munder&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Here, &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are measurable functions, &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;ess&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130417"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978789","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":"Multiplicity of solutions to degenerate elliptic problems involving Grushin operator","authors":"Somnath Gandal ,&nbsp;Jagmohan Tyagi","doi":"10.1016/j.jmaa.2025.130363","DOIUrl":"10.1016/j.jmaa.2025.130363","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we prove a multiplicity result for solutions to the following class of problems&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt; in &lt;/mtext&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt; in &lt;/mtext&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt; on &lt;/mtext&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Here, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the Grushin operator, &lt;span&gt;&lt;math&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is the critical Sobolev exponent, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and Ω is a bounded domain of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; intersecting the degeneration set of the operator. We employ Lusternik-Schnirelman category theorem to prove the result. More precisely, we show that there exists &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;&gt;&lt;/mo&gt;","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130363"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145852490","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":"Dirichlet's series associated with some power series","authors":"Ahmed Sebbar ,&nbsp;Roger Gay","doi":"10.1016/j.jmaa.2026.130408","DOIUrl":"10.1016/j.jmaa.2026.130408","url":null,"abstract":"<div><div>We study the singularities of the Dirichlet series<span><span><span><math><mi>D</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></munder><mfrac><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></mfrac></math></span></span></span> when the power series <span><math><mi>H</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> has a singularity of general type at the point <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>. This extends a recent result of L.M. Navas, J. Ruiz, and J.L. Varona, and connects to foundational ideas explored by Hardy, Fekete, and others. The tools employed include classical methods from the theory of Dirichlet series, particularly those often used in connection with the Riemann zeta function, namely the Mellin transform and splitting methods. These techniques were also used by Navas, Ruiz, and Varona. The polylogarithm function plays a fundamental role in this work.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130408"},"PeriodicalIF":1.2,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023455","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}