Pub Date : 2026-01-01Epub Date: 2025-09-11DOI: 10.1016/j.na.2025.113940
R. Dhanya , Jacques Giacomoni , Ritabrata Jana
In this article, we examine the Hölder regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator with singular data, under the minimal assumption that . The regularity result is twofold: we establish interior gradient Hölder regularity for locally bounded data and boundary regularity for singular data. We prove both boundary Hölder and boundary gradient Hölder regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.
{"title":"Interior and boundary regularity of mixed local nonlocal problem with singular data and its applications","authors":"R. Dhanya , Jacques Giacomoni , Ritabrata Jana","doi":"10.1016/j.na.2025.113940","DOIUrl":"10.1016/j.na.2025.113940","url":null,"abstract":"<div><div>In this article, we examine the Hölder regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator <span><math><mrow><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><msubsup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></math></span> with singular data, under the minimal assumption that <span><math><mrow><mi>p</mi><mo>></mo><mi>s</mi><mi>q</mi></mrow></math></span>. The regularity result is twofold: we establish interior gradient Hölder regularity for locally bounded data and boundary regularity for singular data. We prove both boundary Hölder and boundary gradient Hölder regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113940"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048353","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}
-type foliations are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping with the Bott connection we consider the scalar horizontal curvature as well as a new local invariant induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of and . The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of -type foliations allows us to consider the pull-back of Korányi balls to . We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when is locally isometric as a sub-Riemannian manifold to its -type tangent group.
{"title":"Local invariants and geometry of the sub-Laplacian on H-type foliations","authors":"Wolfram Bauer , Irina Markina , Abdellah Laaroussi , Sylvie Vega-Molino","doi":"10.1016/j.na.2025.113934","DOIUrl":"10.1016/j.na.2025.113934","url":null,"abstract":"<div><div><span><math><mi>H</mi></math></span>-type foliations <span><math><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>)</mo></mrow></math></span> are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping <span><math><mi>M</mi></math></span> with the Bott connection we consider the scalar horizontal curvature <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> as well as a new local invariant <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub></math></span> induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of <span><math><msub><mrow><mi>κ</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>V</mi></mrow></msub></math></span>. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of <span><math><mi>H</mi></math></span>-type foliations allows us to consider the pull-back of Korányi balls to <span><math><mi>M</mi></math></span>. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when <span><math><mi>M</mi></math></span> is locally isometric as a sub-Riemannian manifold to its <span><math><mi>H</mi></math></span>-type tangent group.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113934"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144920377","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 : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.na.2025.113939
Tongkeun Chang, Kyungkeun Kang
Local behavior near the boundary is analyzed for solutions of the Stokes and Navier–Stokes equations in the half space with localized non-smooth boundary data. We construct solutions to the Stokes equations whose velocity fields are unbounded near the boundary away from the support of boundary data, although the velocity and its gradient of solutions are locally square integrable. This is an improvement compared to known results in the sense that the velocity field itself is unbounded, since previously constructed solutions were bounded near the boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to near the boundary for . For such examples, the corresponding pressures turn out not to be locally integrable. A similar construction, via a perturbation argument, is available to the Navier–Stokes equations near the boundary as well.
{"title":"Singular velocity of the Stokes and Navier–Stokes equations near boundary in the half-space","authors":"Tongkeun Chang, Kyungkeun Kang","doi":"10.1016/j.na.2025.113939","DOIUrl":"10.1016/j.na.2025.113939","url":null,"abstract":"<div><div>Local behavior near the boundary is analyzed for solutions of the Stokes and Navier–Stokes equations in the half space with localized non-smooth boundary data. We construct solutions to the Stokes equations whose velocity fields are unbounded near the boundary away from the support of boundary data, although the velocity and its gradient of solutions are locally square integrable. This is an improvement compared to known results in the sense that the velocity field itself is unbounded, since previously constructed solutions were bounded near the boundary, although their normal derivatives are singular. We also establish singular solutions and their derivatives that do not belong to <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi></mrow></msubsup></math></span> near the boundary for <span><math><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></math></span>. For such examples, the corresponding pressures turn out not to be locally integrable. A similar construction, via a perturbation argument, is available to the Navier–Stokes equations near the boundary as well.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113939"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932826","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 : 2026-01-01Epub Date: 2025-08-11DOI: 10.1016/j.na.2025.113913
L. Ambrosio, F. Renzi, F. Vitillaro
We prove that every one-dimensional locally normal metric current, intended in the sense of U. Lang and S. Wenger, admits a nice integral representation through currents associated to (possibly unbounded) curves with locally finite length, generalizing the result shown by E. Paolini and E. Stepanov in the special case of Ambrosio–Kirchheim normal currents. Our result holds in Polish spaces, or more generally in complete metric spaces for 1-currents with tight support.
{"title":"The superposition principle for local 1-dimensional currents","authors":"L. Ambrosio, F. Renzi, F. Vitillaro","doi":"10.1016/j.na.2025.113913","DOIUrl":"10.1016/j.na.2025.113913","url":null,"abstract":"<div><div>We prove that every one-dimensional locally normal metric current, intended in the sense of U. Lang and S. Wenger, admits a nice integral representation through currents associated to (possibly unbounded) curves with locally finite length, generalizing the result shown by E. Paolini and E. Stepanov in the special case of Ambrosio–Kirchheim normal currents. Our result holds in Polish spaces, or more generally in complete metric spaces for 1-currents with tight support.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113913"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809686","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 : 2026-01-01Epub Date: 2025-08-06DOI: 10.1016/j.na.2025.113905
Gustavo de Paula Ramos
We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami et al. (2022) and the strategy used to avoid dichotomy in Bellazzini and Siciliano (2011). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schrödinger–Poisson system.
{"title":"Minimizers of mass-constrained functionals involving a nonattractive point interaction","authors":"Gustavo de Paula Ramos","doi":"10.1016/j.na.2025.113905","DOIUrl":"10.1016/j.na.2025.113905","url":null,"abstract":"<div><div>We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami et al. (2022) and the strategy used to avoid dichotomy in Bellazzini and Siciliano (2011). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schrödinger–Poisson system.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113905"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779748","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 : 2026-01-01Epub Date: 2025-08-26DOI: 10.1016/j.na.2025.113930
Thomas Hamori , Changhui Tan
We present a new family of second-order traffic flow models, extending the Aw–Rascle–Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models.
In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models.
Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.
{"title":"On the Aw–Rascle–Zhang traffic models with nonlocal look-ahead interactions","authors":"Thomas Hamori , Changhui Tan","doi":"10.1016/j.na.2025.113930","DOIUrl":"10.1016/j.na.2025.113930","url":null,"abstract":"<div><div>We present a new family of second-order traffic flow models, extending the Aw–Rascle–Zhang (ARZ) model to incorporate nonlocal interactions. Our model includes a specific nonlocal Arrhenius-type look-ahead slowdown factor. We establish both local and global well-posedness theories for these nonlocal ARZ models.</div><div>In contrast to the local ARZ model, where generic smooth initial data typically lead to finite-time shock formation, we show that our nonlocal ARZ model exhibits global regularity for a class of smooth subcritical initial data. Our result highlights the potential of nonlocal interactions to mitigate shock formations in second-order traffic flow models.</div><div>Our analytical approach relies on investigating phase plane dynamics. We introduce a novel comparison principle based on a mediant inequality to effectively handle the nonlocal information inherent in our model.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113930"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895313","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 : 2026-01-01Epub Date: 2025-09-16DOI: 10.1016/j.na.2025.113914
A.B. Lima , R.M. Batista , P.A. Sousa
We study the -stability of hypersurfaces with null expansion in an -dimensional initial data set with cosmological constant . First, under natural energy conditions, we demonstrate that admits a metric with positive scalar curvature. Second, for a -stable surface of genus , we establish an inequality relating the area of , its genus, , and the charge . Moreover, if equality holds and , is a round 2-sphere. Finally, for a -stable, two-sided, closed hypersurface in a 5-dimensional initial data set satisfying natural energy conditions, we derive an inequality involving the area of , its charge , and a positive constant depending on the total traceless Ricci curvature of . Equality implies that is isometric to .
{"title":"Some results on g-stability for hypersurfaces in an initial data set","authors":"A.B. Lima , R.M. Batista , P.A. Sousa","doi":"10.1016/j.na.2025.113914","DOIUrl":"10.1016/j.na.2025.113914","url":null,"abstract":"<div><div>We study the <span><math><mi>g</mi></math></span>-stability of hypersurfaces <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> with null expansion <span><math><mrow><msup><mrow><mi>θ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi>h</mi><mo>≥</mo><mn>0</mn></mrow></math></span> in an <span><math><mi>n</mi></math></span>-dimensional initial data set <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with cosmological constant <span><math><mi>Λ</mi></math></span>. First, under natural energy conditions, we demonstrate that <span><math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⊂</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> admits a metric with positive scalar curvature. Second, for a <span><math><mi>g</mi></math></span>-stable surface <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of genus <span><math><mrow><mi>g</mi><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow></mrow></math></span>, we establish an inequality relating the area of <span><math><mi>Σ</mi></math></span>, its genus, <span><math><mi>Λ</mi></math></span>, and the charge <span><math><mrow><mi>q</mi><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow></mrow></math></span>. Moreover, if equality holds and <span><math><mrow><mi>Λ</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a round 2-sphere. Finally, for a <span><math><mi>g</mi></math></span>-stable, two-sided, closed hypersurface <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> in a 5-dimensional initial data set <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> satisfying natural energy conditions, we derive an inequality involving the area of <span><math><mi>Σ</mi></math></span>, its charge <span><math><mrow><mi>q</mi><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow></mrow></math></span>, and a positive constant depending on the total traceless Ricci curvature of <span><math><mi>Σ</mi></math></span>. Equality implies that <span><math><msup><mrow><mi>Σ</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> is isometric to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113914"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094804","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 : 2026-01-01Epub Date: 2025-08-06DOI: 10.1016/j.na.2025.113912
Lichun Liang
In this paper, we investigate the asymptotic behavior of viscosity solutions for Monge–Ampère equations in the half space with a Dirichlet boundary condition on the flat boundary. Via the Kelvin transform, we characterize the asymptotic remainders by a single function near the origin. Such a function is smooth in the neighborhood of the origin in even dimension, but only in odd dimension.
{"title":"Higher order expansion at infinity of solutions for Monge–Ampère equations in the half space","authors":"Lichun Liang","doi":"10.1016/j.na.2025.113912","DOIUrl":"10.1016/j.na.2025.113912","url":null,"abstract":"<div><div>In this paper, we investigate the asymptotic behavior of viscosity solutions for Monge–Ampère equations in the half space with a Dirichlet boundary condition on the flat boundary. Via the Kelvin transform, we characterize the asymptotic remainders by a single function near the origin. Such a function is smooth in the neighborhood of the origin in even dimension, but only <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> <span><math><mrow><mo>(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn><mo>)</mo></mrow></math></span> in odd dimension.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113912"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779750","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 : 2026-01-01Epub Date: 2025-08-14DOI: 10.1016/j.na.2025.113918
Jacek Dziubański
Let be a Schrödinger operator on , where , . We give a short proof of dimension free -estimates, , , for the vector of the Riesz transforms The constant in the estimates does not depend on the potential . We simultaneously provide a short proof of the weak-type estimates for .
{"title":"On dimension-free and potential-free estimates for Riesz transforms associated with Schrödinger operators","authors":"Jacek Dziubański","doi":"10.1016/j.na.2025.113918","DOIUrl":"10.1016/j.na.2025.113918","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>L</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> be a Schrödinger operator on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, where <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>V</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. We give a short proof of dimension free <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>-estimates, <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, for the vector of the Riesz transforms <span><math><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>∂</mi></mrow><mrow><mi>∂</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo><mfrac><mrow><mi>∂</mi></mrow><mrow><mi>∂</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><mfrac><mrow><mi>∂</mi></mrow><mrow><mi>∂</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math></span> The constant in the estimates does not depend on the potential <span><math><mi>V</mi></math></span>. We simultaneously provide a short proof of the weak-type <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> estimates for <span><math><mrow><mfrac><mrow><mi>∂</mi></mrow><mrow><mi>∂</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113918"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144842243","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 : 2026-01-01Epub Date: 2025-09-24DOI: 10.1016/j.na.2025.113945
Daniel Devine , Paschalis Karageorgis
When it comes to the nonlinear heat equation , the stability of positive radial steady states in the supercritical case was established in the classical paper by Gui, Ni and Wang. We extend this result to systems of reaction–diffusion equations by studying the positive radial steady states of the parabolic Hénon–Lane–Emden system where , and . Assume that lies either on or above the Joseph–Lundgren critical curve which arose in the work of Chen, Dupaigne and Ghergu. Then all positive radial steady states have the same asymptotic behavior at infinity, and they are all stable solutions of the parabolic Hénon–Lane–Emden system in .
对于非线性热方程ut−Δu=up， Gui、Ni和Wang在经典论文中建立了超临界情况下径向正稳态的稳定性。通过研究抛物型h - lane - emden系统ut−Δu=|x| kvpinrnx(0,∞),vt−Δv=|x| luqinrnx(0,∞)，其中k，l≥0,p,q≥1,pq>；1的正径向稳态，我们将这一结果推广到反应扩散方程系统。假设（p,q）位于Joseph-Lundgren临界曲线上或之上，该曲线由Chen、Dupaigne和Ghergu提出。那么所有正径向稳态在无穷远处都具有相同的渐近性质，它们都是Rn中抛物型h - lane - emden系统的稳定解。
{"title":"Stability of positive radial steady states for the parabolic Hénon–Lane–Emden system","authors":"Daniel Devine , Paschalis Karageorgis","doi":"10.1016/j.na.2025.113945","DOIUrl":"10.1016/j.na.2025.113945","url":null,"abstract":"<div><div>When it comes to the nonlinear heat equation <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></math></span>, the stability of positive radial steady states in the supercritical case was established in the classical paper by Gui, Ni and Wang. We extend this result to systems of reaction–diffusion equations by studying the positive radial steady states of the parabolic Hénon–Lane–Emden system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>u</mi></mtd><mtd><mo>=</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mi>v</mi></mrow><mrow><mi>p</mi></mrow></msup></mtd><mtd><mtext>in</mtext><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><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>v</mi></mtd><mtd><mo>=</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>l</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup></mtd><mtd><mtext>in</mtext><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><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mi>q</mi><mo>></mo><mn>1</mn></mrow></math></span>. Assume that <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span> lies either on or above the Joseph–Lundgren critical curve which arose in the work of Chen, Dupaigne and Ghergu. Then all positive radial steady states have the same asymptotic behavior at infinity, and they are all stable solutions of the parabolic Hénon–Lane–Emden system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"262 ","pages":"Article 113945"},"PeriodicalIF":1.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157797","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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