Illya M. Karabash , Christina Lienstromberg , Juan J.L. Velázquez
{"title":"Multi-parameter Hopf bifurcations of rimming flows","authors":"Illya M. Karabash , Christina Lienstromberg , Juan J.L. Velázquez","doi":"10.1016/j.jde.2025.02.053","DOIUrl":"10.1016/j.jde.2025.02.053","url":null,"abstract":"<div><div>In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of Benilov, Kopteva, O'Brien (2005) <span><span>[7]</span></span>, which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic in time solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113182"},"PeriodicalIF":2.4,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sufficient conditions for local Lebesgue solvability of canceling and elliptic linear differential equations with measure data","authors":"V. Biliatto, T. Picon","doi":"10.1016/j.jde.2025.02.050","DOIUrl":"10.1016/j.jde.2025.02.050","url":null,"abstract":"<div><div>We present sufficient conditions for the local Lebesgue solvability of the equation <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mo>)</mo><mi>f</mi><mo>=</mo><mi>μ</mi></math></span> with Borel measure data <em>μ</em>, associated to an elliptic linear differential operator <span><math><mi>A</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mo>)</mo></math></span> of order <em>m</em> with smooth complex variable coefficients. Our method for obtaining local solutions <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> with <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> assumes that the measure has finite strong <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>-energy. The solvability for the endpoint case <span><math><mi>p</mi><mo>=</mo><mo>∞</mo></math></span> is studied in the setting of elliptic and canceling operators as a consequence of new local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> estimates on measures for a special class of vector fields.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113179"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Response tori in Hamiltonian systems of high order degeneracy - the super-critical case","authors":"Lu Xu , Wen Si , Yingfei Yi","doi":"10.1016/j.jde.2025.02.051","DOIUrl":"10.1016/j.jde.2025.02.051","url":null,"abstract":"<div><div>Consider the quasi-periodically forced, 2nd order differential equations<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>+</mo><mi>λ</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>+</mo><mi>ε</mi><mi>f</mi><mo>(</mo><mi>ω</mi><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>λ</mi><mo>≠</mo><mn>0</mn></math></span> is a constant, <span><math><mi>l</mi><mo>≥</mo><mn>2</mn></math></span> is an integer, <span><math><mi>ω</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is a Diophantine frequency vector, <em>ε</em> is a small parameter, and <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> is real analytic. It is shown in <span><span>[18]</span></span> that if the leading order <em>p</em> of non-degeneracy of <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> satisfies <span><math><mn>0</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>l</mi><mo>/</mo><mn>2</mn></math></span>, then response solutions of the equation exist under some minor conditions. Indeed, <span><math><mi>l</mi><mo>/</mo><mn>2</mn></math></span> is the critical order of non-degeneracy of <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> such that relative equilibria of the equation can be solved from its averaged equation - a typical mechanism for the existence of response solutions in perturbed, quasi-periodically forced, 2nd order nonlinear equations. In this paper, we consider the existence of response solutions of the equation for the super-critical case, i.e., <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> is degenerate at least up to an order <span><math><mi>p</mi><mo>≥</mo><mi>l</mi><mo>/</mo><mn>2</mn></math></span>. We will show in this case that response solutions can still exist around perturbed relative equilibria of the normalized equation by considering non-degeneracy of the new perturbation after the normalization that is of at least <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> order. This reveals a mechanism for the existence of response solutions of the equation in the super-critical case.</div><div>For the sake of generality, we will actually consider a general Hamiltonian normal form containing the normalized equation as a particular case. We will prove a general theorem concerning the existence of response tori of the normal form through averaging, finding rela","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 612-645"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky
{"title":"The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone","authors":"Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky","doi":"10.1016/j.jde.2025.02.016","DOIUrl":"10.1016/j.jde.2025.02.016","url":null,"abstract":"<div><div>In a recent paper, we presented scenarios of long-time asymptotics for the solution of the focusing nonlinear Schrödinger equation with initial data approaching plane waves of the form <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>i</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi></mrow></msup></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>i</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>x</mi></mrow></msup></math></span> at minus and plus infinity, respectively. In the shock case <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> some scenarios include sectors of genus 3, that is, sectors <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mi>ξ</mi><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mi>ξ</mi><mo>≔</mo><mi>x</mi><mo>/</mo><mi>t</mi></math></span>, where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface of genus 3. The present paper deals with the asymptotic analysis in a transition zone between two genus 3 sectors. The leading term is expressed in terms of elliptic functions attached to a Riemann surface of genus 1. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 747-801"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: A Green function approach","authors":"Gabriele Grillo, Dario D. Monticelli, Fabio Punzo","doi":"10.1016/j.jde.2025.02.062","DOIUrl":"10.1016/j.jde.2025.02.062","url":null,"abstract":"<div><div>We consider the porous medium equation (PME) on complete noncompact manifolds <em>M</em> of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space <em>X</em> of functions, strictly larger than <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, in which the Green function on <em>M</em> appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is nonnegative and belongs to <em>X</em>. Smoothing estimates are also proved to hold both for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to <em>X</em> as well.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113191"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tomasz Klimsiak , Tomasz Komorowski , Lorenzo Marino
{"title":"Homogenization of stable-like operators with random, ergodic coefficients","authors":"Tomasz Klimsiak , Tomasz Komorowski , Lorenzo Marino","doi":"10.1016/j.jde.2025.02.054","DOIUrl":"10.1016/j.jde.2025.02.054","url":null,"abstract":"<div><div>We show homogenization for a family of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-valued stable-like processes <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>ε</mi><mo>;</mo><mi>θ</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, <span><math><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, whose (random) Fourier symbols equal <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>ε</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mi>q</mi><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mi>ε</mi></mrow></mfrac><mo>,</mo><mi>ε</mi><mi>ξ</mi><mo>;</mo><mi>θ</mi><mo>)</mo></math></span>, where<span><span><span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mo>=</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></munder><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>z</mi><mo>⋅</mo><mi>ξ</mi></mrow></msup><mo>+</mo><mi>i</mi><mi>z</mi><mo>⋅</mo><mi>ξ</mi><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo>≤</mo><mn>1</mn><mo>}</mo></mrow></msub><mo>)</mo></mrow><mspace></mspace><mfrac><mrow><mi>a</mi><mo>(</mo><mi>x</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mi>z</mi><mo>⋅</mo><mi>z</mi></mrow><mrow><mo>|</mo><mi>z</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn><mo>+</mo><mi>α</mi></mrow></msup></mrow></mfrac><mspace></mspace><mi>d</mi><mi>z</mi><mo>,</mo></math></span></span></span> for <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mi>Θ</mi></math></span>. Here <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and the family <span><math><msub><mrow><mo>(</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></msub></math></span> of <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space <figure><img></figure>. We assume that the random field is deterministically bounded and non-degenerate, i.e. <span><math><mo>|</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>Λ</mi></math></span> and <span><math><mtext>Tr</mtext><mo>(</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>;</mo><mi>θ</mi><mo>)</mo><mo>)</mo><mo>≥</mo><mi>λ</mi></math></span> for some <span><math><mi>Λ</mi><mo>,</mo><mi>λ</mi><mo>></mo><mn>0</mn></math></span> an","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113183"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The derivative structure for a quadratic nonlinearity and uniqueness for SQG","authors":"Tsukasa Iwabuchi","doi":"10.1016/j.jde.2025.02.057","DOIUrl":"10.1016/j.jde.2025.02.057","url":null,"abstract":"<div><div>We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, we demonstrate that the uniqueness of the mild solution holds in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivatives for the nonlinear term involving fractional derivatives of the Dirichlet Laplacian.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 802-825"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The local well-posedness of analytic solution to the boundary layer system for compressible flow in three dimensions","authors":"Yufeng Chen , Lizhi Ruan , Anita Yang","doi":"10.1016/j.jde.2025.02.056","DOIUrl":"10.1016/j.jde.2025.02.056","url":null,"abstract":"<div><div>In this paper, we consider three dimensional boundary layer equations for compressible isentropic flow with no-slip boundary condition. The local well-posedness of the compressible boundary layer system is established when the initial datum is real-analytic in the tangential direction and has Sobolev regularity in the normal direction. The proof is based on the introduction of a change of variables to eliminate the linear growth in normal direction and subtle energy estimates with algebraic weights.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 716-746"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New type of solutions for the critical polyharmonic equation","authors":"Wenjing Chen, Zexi Wang","doi":"10.1016/j.jde.2025.02.058","DOIUrl":"10.1016/j.jde.2025.02.058","url":null,"abstract":"<div><div>In this paper, we consider the following critical polyharmonic equation<span><span><span><math><mrow><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><msup><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>y</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>3</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><msup><mrow><mi>m</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></math></span>, <span><math><mi>N</mi><mo>></mo><mn>4</mn><mi>m</mi><mo>+</mo><mn>1</mn></math></span>, <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, and <span><math><mi>V</mi><mo>(</mo><mo>|</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo></math></span> is a bounded nonnegative function in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>3</mn></mrow></msup></math></span>. By using the reduction argument and local Pohoz̆aev identities, we prove that if <span><math><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>)</mo></math></span> has a stable critical point <span><math><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>)</mo></math></span> with <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>″</mo></mrow></msubsup><mo>)</mo><mo>></mo><mn>0</mn></math></span>, then the above problem has a new type of solutions, which concentrate at points lying on the top and the bottom circles of a cylinder.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 678-715"},"PeriodicalIF":2.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On existence of normalized solutions to some classes of elliptic problems with L2-supercritical growth","authors":"Claudianor O. Alves , Liejun Shen","doi":"10.1016/j.jde.2025.02.059","DOIUrl":"10.1016/j.jde.2025.02.059","url":null,"abstract":"<div><div>In this paper, we present a new approach that can be used to prove the existence of normalized solution for elliptic problems with nonlinearity having an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-supercritical growth, where the domain can be bounded, the whole <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> or the Half space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span>. Moreover, this method also makes the studies of normalized solutions for problem involving magnetic field available.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113188"},"PeriodicalIF":2.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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