Pub Date : 2025-02-19DOI: 10.1016/j.na.2025.113776
Alba Lia Masiello , Francesco Salerno
In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya–Szegő type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya–Szegő inequality for the -Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso (1989) for solutions to the -Hessian equation.
As an application of the first result, we prove a quantitative version of the Faber–Krahn and Saint-Venant inequalities for these equations.
{"title":"A quantitative result for the k-Hessian equation","authors":"Alba Lia Masiello , Francesco Salerno","doi":"10.1016/j.na.2025.113776","DOIUrl":"10.1016/j.na.2025.113776","url":null,"abstract":"<div><div>In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya–Szegő type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya–Szegő inequality for the <span><math><mi>k</mi></math></span>-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso (1989) for solutions to the <span><math><mi>k</mi></math></span>-Hessian equation.</div><div>As an application of the first result, we prove a quantitative version of the Faber–Krahn and Saint-Venant inequalities for these equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113776"},"PeriodicalIF":1.3,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437533","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}
Pub Date : 2025-02-18DOI: 10.1016/j.na.2025.113775
Lihua Dong
This paper is concerned with asymptotic stability of certain stationary solution to the 3D Boussinesq equations in the whole space with a damping term in the velocity equation. Precisely, the decay rates of solutions is optimal in sense that these rates coincide with that of the linearized equations.
{"title":"The optimal decay rates for solutions to the 3D Boussinesq equations with a velocity damping term in R3","authors":"Lihua Dong","doi":"10.1016/j.na.2025.113775","DOIUrl":"10.1016/j.na.2025.113775","url":null,"abstract":"<div><div>This paper is concerned with asymptotic stability of certain stationary solution to the 3D Boussinesq equations in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with a damping term in the velocity equation. Precisely, the decay rates of solutions is optimal in sense that these rates coincide with that of the linearized equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113775"},"PeriodicalIF":1.3,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429538","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}
Pub Date : 2025-02-16DOI: 10.1016/j.na.2025.113753
Y. Chitour , M. Kafnemer , P. Martinez , B. Mebkhout
In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the framework, with .
We begin by addressing the well-posedness problem, establishing the existence and uniqueness of weak and strong solutions for , under suitable assumptions on the damping function.
Next, we study the asymptotic behaviour of the associated energy when , and we provide decay estimates that appear to be almost optimal compared to similar problems with boundary damping.
Our work is motivated by earlier studies, particularly, those by Chitour, Marx and Prieur (2020), and Haraux (1978). The proofs combine arguments from Kafnemer, Mebkhout and Chitour (2022) for wave equation in the framework with a linear damping, techniques of weighted energy estimates introduced in Martinez (1999), new integral inequalities for , and convex analysis tools when .
{"title":"Lp asymptotic stability of 1D damped wave equation with nonlinear damping","authors":"Y. Chitour , M. Kafnemer , P. Martinez , B. Mebkhout","doi":"10.1016/j.na.2025.113753","DOIUrl":"10.1016/j.na.2025.113753","url":null,"abstract":"<div><div>In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> framework, with <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>.</div><div>We begin by addressing the well-posedness problem, establishing the existence and uniqueness of weak and strong solutions for <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, under suitable assumptions on the damping function.</div><div>Next, we study the asymptotic behaviour of the associated energy when <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, and we provide decay estimates that appear to be almost optimal compared to similar problems with boundary damping.</div><div>Our work is motivated by earlier studies, particularly, those by Chitour, Marx and Prieur (2020), and Haraux (1978). The proofs combine arguments from Kafnemer, Mebkhout and Chitour (2022) for wave equation in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> framework with a linear damping, techniques of weighted energy estimates introduced in Martinez (1999), new integral inequalities for <span><math><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow></math></span>, and convex analysis tools when <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113753"},"PeriodicalIF":1.3,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143421249","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}
Pub Date : 2025-02-12DOI: 10.1016/j.na.2025.113774
Cosmin Burtea , Maja Szlenk
We prove the existence of weak solutions for the steady Navier–Stokes system for compressible non-Newtonian fluids on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power and the pressure is given by , we construct a solution provided that and is sufficiently large, depending on the values of . Additionally, we also show the existence for time-discretized model for Herschel–Bulkley fluids, where the viscosity has a singular part.
{"title":"Weak solutions to the Navier–Stokes equations for steady compressible non-Newtonian fluids","authors":"Cosmin Burtea , Maja Szlenk","doi":"10.1016/j.na.2025.113774","DOIUrl":"10.1016/j.na.2025.113774","url":null,"abstract":"<div><div>We prove the existence of weak solutions for the steady Navier–Stokes system for compressible non-Newtonian fluids on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power <span><math><mi>r</mi></math></span> and the pressure is given by <span><math><msup><mrow><mi>ϱ</mi></mrow><mrow><mi>γ</mi></mrow></msup></math></span>, we construct a solution provided that <span><math><mrow><mi>r</mi><mo>></mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mi>γ</mi></math></span> is sufficiently large, depending on the values of <span><math><mi>r</mi></math></span>. Additionally, we also show the existence for time-discretized model for Herschel–Bulkley fluids, where the viscosity has a singular part.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113774"},"PeriodicalIF":1.3,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388285","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}
Pub Date : 2025-02-11DOI: 10.1016/j.na.2025.113773
Te Ba , Chao Zheng
Generalized hyperbolic circle packings were introduced in Ba et al. (2023) as the generalization of tangential circle packings in hyperbolic background geometry. To find generalized hyperbolic circle packings on surfaces with prescribed total geodesic curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial th Calabi flow for generalized hyperbolic circle packings on surfaces. We establish several equivalent conditions regarding the longtime behaviors of these combinatorial curvature flows. This provides effective algorithms for finding the generalized hyperbolic circle packings with prescribed total geodesic curvatures on surfaces.
{"title":"Combinatorial curvature flows for generalized hyperbolic circle packings on surfaces","authors":"Te Ba , Chao Zheng","doi":"10.1016/j.na.2025.113773","DOIUrl":"10.1016/j.na.2025.113773","url":null,"abstract":"<div><div>Generalized hyperbolic circle packings were introduced in Ba et al. (2023) as the generalization of tangential circle packings in hyperbolic background geometry. To find generalized hyperbolic circle packings on surfaces with prescribed total geodesic curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial <span><math><mi>p</mi></math></span>th Calabi flow for generalized hyperbolic circle packings on surfaces. We establish several equivalent conditions regarding the longtime behaviors of these combinatorial curvature flows. This provides effective algorithms for finding the generalized hyperbolic circle packings with prescribed total geodesic curvatures on surfaces.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113773"},"PeriodicalIF":1.3,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378782","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}
Pub Date : 2025-02-08DOI: 10.1016/j.na.2025.113772
Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez
<div><div>We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term <span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></mrow></math></span> posed for <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mspace></mspace><mn>3</mn></mrow></math></span>, in the sub-critical range of the fast diffusion equation <span><math><mrow><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>N</mi></mrow></math></span>. We consider <span><math><mrow><mi>σ</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mo>max</mo><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mo><</mo><mi>p</mi><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>m</mi><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mi>σ</mi><mo>)</mo></mrow></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mspace></mspace><msub><mrow><mi>p</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>m</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>.</mo></mrow></math></span> We show that, on the one hand, positive self-similar solutions at any time <span><math><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math></span>, in the form <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>f</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mi>f</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∼</mo><mi>C</mi><msup><mrow><mi>ξ</mi></mrow><mrow><mo>−</mo><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>m</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn
{"title":"Extinction and non-extinction profiles for the sub-critical fast diffusion equation with weighted source","authors":"Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez","doi":"10.1016/j.na.2025.113772","DOIUrl":"10.1016/j.na.2025.113772","url":null,"abstract":"<div><div>We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term <span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>σ</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></mrow></math></span> posed for <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mspace></mspace><mn>3</mn></mrow></math></span>, in the sub-critical range of the fast diffusion equation <span><math><mrow><mn>0</mn><mo><</mo><mi>m</mi><mo><</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>N</mi></mrow></math></span>. We consider <span><math><mrow><mi>σ</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mo>max</mo><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mo><</mo><mi>p</mi><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>m</mi><mrow><mo>(</mo><mi>N</mi><mo>+</mo><mi>σ</mi><mo>)</mo></mrow></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mspace></mspace><msub><mrow><mi>p</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>σ</mi><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>m</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>.</mo></mrow></math></span> We show that, on the one hand, positive self-similar solutions at any time <span><math><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math></span>, in the form <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msup><mi>f</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mi>f</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>∼</mo><mi>C</mi><msup><mrow><mi>ξ</mi></mrow><mrow><mo>−</mo><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>m</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113772"},"PeriodicalIF":1.3,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349453","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}
Pub Date : 2025-02-06DOI: 10.1016/j.na.2025.113762
Anna Maria Candela , Kanishka Perera , Addolorata Salvatore
<div><div>The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically “linear” problem <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mo>div</mo><mfenced><mrow><mfenced><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msup></mrow></mfenced><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>+</mo><mi>s</mi><mspace></mspace><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace></mspace><mspace></mspace><mspace></mspace><mo>=</mo><mspace></mspace><mi>μ</mi><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext><mi>Ω</mi><mtext>,</mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mtext>on</mtext><mi>∂</mi><mi>Ω</mi><mtext>,</mtext></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>></mo><mn>1</mn><mo>/</mo><mi>p</mi></mrow></math></span>, both the coefficients <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> and far away from 0, <span><math><mrow><mi>μ</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, and the “perturbation” term <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is a Carathéodory function on <span><math><mrow><mi>Ω</mi><mo>×</mo><mi>R</mi></mrow></math></span> which grows as <span><math><msup><mrow><mrow><mo>|</mo><mi>t</mi><mo>|</mo></mrow></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mrow><mn>1</m
{"title":"Existence results for a borderline case of a class of p-Laplacian problems","authors":"Anna Maria Candela , Kanishka Perera , Addolorata Salvatore","doi":"10.1016/j.na.2025.113762","DOIUrl":"10.1016/j.na.2025.113762","url":null,"abstract":"<div><div>The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically “linear” problem <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mo>div</mo><mfenced><mrow><mfenced><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msup></mrow></mfenced><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>+</mo><mi>s</mi><mspace></mspace><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace></mspace><mspace></mspace><mspace></mspace><mo>=</mo><mspace></mspace><mi>μ</mi><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext><mi>Ω</mi><mtext>,</mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mtext>on</mtext><mi>∂</mi><mi>Ω</mi><mtext>,</mtext></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>></mo><mn>1</mn><mo>/</mo><mi>p</mi></mrow></math></span>, both the coefficients <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> and far away from 0, <span><math><mrow><mi>μ</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, and the “perturbation” term <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> is a Carathéodory function on <span><math><mrow><mi>Ω</mi><mo>×</mo><mi>R</mi></mrow></math></span> which grows as <span><math><msup><mrow><mrow><mo>|</mo><mi>t</mi><mo>|</mo></mrow></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mrow><mn>1</m","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113762"},"PeriodicalIF":1.3,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265967","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}
Pub Date : 2025-02-05DOI: 10.1016/j.na.2025.113760
Arunima Bhattacharya , Anna Skorobogatova
We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of , under compactly supported variations. We show that for smooth convex functionals, a critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most , for some . We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.
{"title":"Variational integrals on Hessian spaces: Partial regularity for critical points","authors":"Arunima Bhattacharya , Anna Skorobogatova","doi":"10.1016/j.na.2025.113760","DOIUrl":"10.1016/j.na.2025.113760","url":null,"abstract":"<div><div>We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, under compactly supported variations. We show that for smooth convex functionals, a <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>∞</mi></mrow></msup></math></span> critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most <span><math><mrow><mi>n</mi><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, for some <span><math><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span>. We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113760"},"PeriodicalIF":1.3,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143268147","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}
Pub Date : 2025-02-03DOI: 10.1016/j.na.2025.113764
Lintao Liu , Kaimin Teng , Shuai Yuan
We consider -constraint minimizers of the Choquard energy functional with a trapping potential . It is known that positive minimizers exist if and only if the parameter satisfies , where is the unique positive radial solution of in . This paper focuses on the local uniqueness of minimizers by using energy estimates, blow-up analysis and establishing the Pohozăev identity.
{"title":"Local uniqueness of minimizers for Choquard type equations","authors":"Lintao Liu , Kaimin Teng , Shuai Yuan","doi":"10.1016/j.na.2025.113764","DOIUrl":"10.1016/j.na.2025.113764","url":null,"abstract":"<div><div>We consider <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-constraint minimizers of the Choquard energy functional with a trapping potential <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. It is known that positive minimizers exist if and only if the parameter <span><math><mi>a</mi></math></span> satisfies <span><math><mrow><mi>a</mi><mo><</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>≔</mo><msubsup><mrow><mo>‖</mo><mi>Q</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math></span>, where <span><math><mi>Q</mi></math></span> is the unique positive radial solution of <span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>u</mi><mo>−</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. This paper focuses on the local uniqueness of minimizers by using energy estimates, blow-up analysis and establishing the Pohozăev identity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113764"},"PeriodicalIF":1.3,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163859","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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