{"title":"Gradient estimate for solutions to Δbu = 0 and Δbu+aulnu+bu=0 on pseudohermitian manifold","authors":"Shitong Niu, Jun Sun","doi":"10.1016/j.jde.2025.02.045","DOIUrl":"10.1016/j.jde.2025.02.045","url":null,"abstract":"<div><div>In this paper, we will establish fundamental <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> estimate for solutions to <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>u</mi><mo>=</mo><mn>0</mn></math></span> and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> estimates for solutions to a class of nonlinear elliptic equations of the form <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>a</mi><mi>u</mi><mi>ln</mi><mo></mo><mi>u</mi><mo>+</mo><mi>b</mi><mi>u</mi><mo>=</mo><mn>0</mn></math></span> on pseudohermitian manifold. The proof consists of combining the Nash-Moser iteration and Saloff-Coste type Sobolev inequalities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 262-282"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436943","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 Lane-Emden system","authors":"Wenjing Chen, Xiaomeng Huang","doi":"10.1016/j.jde.2025.02.046","DOIUrl":"10.1016/j.jde.2025.02.046","url":null,"abstract":"<div><div>In this paper, we consider the critical Lane-Emden system<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mtd><mtd></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mtd><mtd></mtd></mtr><mtr><mtd><mi>u</mi><mo>,</mo><mi>v</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></span></span></span> where <span><math><mi>N</mi><mo>≥</mo><mn>5</mn></math></span>, <span><math><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac></math></span>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo></math></span> are positive radial potentials. Under suitable conditions on <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo></math></span>, we construct a new family of solutions to this system, which are centred 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 318-391"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445157","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}
Bogdan-Vasile Matioc , Luigi Roberti , Christoph Walker
{"title":"Quasilinear parabolic equations with superlinear nonlinearities in critical spaces","authors":"Bogdan-Vasile Matioc , Luigi Roberti , Christoph Walker","doi":"10.1016/j.jde.2025.02.039","DOIUrl":"10.1016/j.jde.2025.02.039","url":null,"abstract":"<div><div>Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations <span><math><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>A</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>u</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> is established in a certain critical case of strict inclusion <span><math><mrow><mi>dom</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊊</mo><mrow><mi>dom</mi></mrow><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for the domains of the (superlinear) function <span><math><mi>u</mi><mo>↦</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> and the quasilinear part <span><math><mi>u</mi><mo>↦</mo><mi>A</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span>. Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space. The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 283-317"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436839","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":"Dispersive decay for the energy-critical nonlinear Schrödinger equation","authors":"Matthew Kowalski","doi":"10.1016/j.jde.2025.02.040","DOIUrl":"10.1016/j.jde.2025.02.040","url":null,"abstract":"<div><div>We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schrödinger equation in spatial dimensions <span><math><mi>d</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span> for both the initial-value and final-state problems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 392-426"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444637","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":"Global existence versus blow-up for a Hardy-Hénon parabolic equation on arbitrary domains","authors":"Ricardo Castillo , Ricardo Freire , Miguel Loayza","doi":"10.1016/j.jde.2025.02.047","DOIUrl":"10.1016/j.jde.2025.02.047","url":null,"abstract":"<div><div>We are concentrating on the nonlinear parabolic problem described by the equation <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>|</mo><mo>⋅</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> in <span><math><mi>Ω</mi><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></math></span> subject to zero Dirichlet conditions on the boundary ∂Ω, where Ω is a general domain that may be either bounded or unbounded. Here, <span><math><mi>h</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, <span><math><mi>γ</mi><mo>></mo><mo>−</mo><mn>2</mn></math></span>, <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>, and we consider only nonnegative initial data. We have derived new conditions for global existence and blow up in finite time in terms of the behavior of the heat semigroup. Our results are particularly relevant when <span><math><mi>γ</mi><mo>=</mo><mn>0</mn></math></span>, as they align with Meier's findings in Meier (1990) <span><span>[29]</span></span>. When <span><math><mi>γ</mi><mo>≠</mo><mn>0</mn></math></span>, our results provide new Fujita exponents.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 427-459"},"PeriodicalIF":2.4,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445156","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":"Global smooth radially symmetric solutions to a multidimensional radiation hydrodynamics model","authors":"Huijiang Zhao , Boran Zhu","doi":"10.1016/j.jde.2025.02.044","DOIUrl":"10.1016/j.jde.2025.02.044","url":null,"abstract":"<div><div>The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although solutions of the compressible Euler system will generally develop singularity no matter how smooth and small the initial data are, it is believed that the radiation effect does imply some dissipative mechanism, which can guarantee the global regularity of the solutions of the radiative Euler equations at least for small initial data.</div><div>Such an expectation was rigorously justified for the one-dimensional case, as for the multidimensional case, to the best of our knowledge, no result was available up to now. The main purpose of this paper is to show that the initial-boundary value problem of such a radiative Euler equation in a three-dimensional bounded concentric annular domain does admit a unique global smooth radially symmetric solution provided that the initial data is sufficiently small.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 123-156"},"PeriodicalIF":2.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430296","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":"Global Navier-Stokes flows in intermediate spaces","authors":"Zachary Bradshaw , Misha Chernobai , Tai-Peng Tsai","doi":"10.1016/j.jde.2025.02.025","DOIUrl":"10.1016/j.jde.2025.02.025","url":null,"abstract":"<div><div>We construct global weak solutions of the three dimensional incompressible Navier-Stokes equations in intermediate spaces between the space of uniformly locally square integrable functions and Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge the existence theorems of Lemarié-Rieusset and of Bradshaw, Kukavica and Tsai. An application to eventual regularity is included which generalizes the prior work of Bradshaw, Kukavica and Tsai as well as Bradshaw, Kukavica and Ozanski.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 50-87"},"PeriodicalIF":2.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430294","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":"Prodi–Serrin condition for 3D MHD equations via one directional derivative of velocity and magnetic fields","authors":"Chenyin Qian , Shiyi Su , Ting Zhang","doi":"10.1016/j.jde.2025.02.033","DOIUrl":"10.1016/j.jde.2025.02.033","url":null,"abstract":"<div><div>In this paper, we consider regularity conditions on weak solutions of 3D MHD equations with viscosity coefficient <em>μ</em> and resistivity coefficient <em>ν</em> being not equal. The main contribution of the present result is to establish the Prodi-Serrin regularity criterion in the case of <span><math><mi>μ</mi><mo>≠</mo><mi>ν</mi></math></span>. More precisely, it shows that the weak solution of the 3D MHD equations is regular if <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mn>3</mn></mrow></msub><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span> and <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mn>3</mn></mrow></msub><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>=</mo><mn>2</mn></math></span>, <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mo>]</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>∞</mo><mo>]</mo></math></span> or <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>∞</mo><mo>]</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mo>]</mo></math></span>. Moreover, if <span><math><mi>μ</mi><mo>=</mo><mi>ν</mi></math></span>, then the range of <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> can be improved to <span><math><mfrac><mrow><mn>3</mn></mrow>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 88-122"},"PeriodicalIF":2.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430295","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":"Local well-posedness to the free boundary problem of incompressible Euler-Poisson-Nernst-Planck system","authors":"Jingchi Huang, Shanmu Li, Zheng-an Yao","doi":"10.1016/j.jde.2025.02.035","DOIUrl":"10.1016/j.jde.2025.02.035","url":null,"abstract":"<div><div>We are concerned with the local well-posedness of three-dimensional incompressible charged fluids bounded by a free-surface. We show that the Euler-Poisson-Nernst-Planck system, wherein the pressure and electrostatic potential vanish along the free boundary, admits the existence of unique strong (in Sobolev spaces) solution in a short time interval. Our proof is founded on a nonlinear approximation system, chosen to preserve the geometric structure, with the aid of tangentially smoothing and Alinhac good unknowns in terms of boundary regularity, our priori estimates do not suffer from the derivative loss phenomenon.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 157-203"},"PeriodicalIF":2.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430297","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":"Normalized solutions to Choquard equation including the critical exponents and a logarithmic perturbation","authors":"Yinbin Deng , Yulin Shi , Xiaolong Yang","doi":"10.1016/j.jde.2025.02.038","DOIUrl":"10.1016/j.jde.2025.02.038","url":null,"abstract":"<div><div>We study the existence of normalized solutions to the following nonlinear Choquard equation<span><span><span>(0.1)</span><span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>λ</mi><mi>u</mi><mo>=</mo><mi>u</mi><mi>log</mi><mo></mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>μ</mi><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>⁎</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span></span></span> under the mass constraint<span><span><span>(0.2)</span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></munder><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> is a constant, <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>N</mi></math></span>, <span><math><mi>μ</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi></mrow></mfrac><mo>≤</mo><mi>p</mi><mo>≤</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>, and the parameter <span><math><mi>λ</mi><mo>∈</mo><mi>R</mi></math></span> appears as a Lagrange multiplier. Under different assumptions on <em>p</em> and <em>c</em>, we first show the existence of the associated global minimizer which must be a ground state solution of <span><span>(0.1)</span></span> with the mass constraint <span><span>(0.2)</span></span> if <span><math><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi></mrow></mfrac><mo>≤</mo><mi>p</mi><mo>≤</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac></math></span>, and then we prove the existence of ground state solution and mountain-pass solution for <span><span>(0.1)</span></span> with the mass constraint <span><span>(0.2)</span></span> if <span><math><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mo><</mo><mi>p</mi><mo>≤</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>, where <span><math><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> is the Hardy-Littlewood-Sobolev upper critical exponent, <span><math><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi></mrow></mfrac></math></span> is the Hardy-Littlewo","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 204-246"},"PeriodicalIF":2.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143436832","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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