Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"Extremal functions for twisted sharp Sobolev inequalities with lower order remainder terms","authors":"Olivier Druet ,&nbsp;Emmanuel Hebey","doi":"10.1016/j.na.2025.113758","DOIUrl":"10.1016/j.na.2025.113758","url":null,"abstract":"<div><div>We prove existence of extremal functions and compactness of the set of extremal functions for twisted sharp 4-dimensional Sobolev inequalities with lower order remainder terms.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113758"},"PeriodicalIF":1.3,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162490","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":"Universal differentiability sets in Laakso space","authors":"Sylvester Eriksson-Bique ,&nbsp;Andrea Pinamonti ,&nbsp;Gareth Speight","doi":"10.1016/j.na.2025.113752","DOIUrl":"10.1016/j.na.2025.113752","url":null,"abstract":"<div><div>We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113752"},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162487","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":"Semiclassical states for the curl–curl problem","authors":"Bartosz Bieganowski ,&nbsp;Adam Konysz ,&nbsp;Jarosław Mederski","doi":"10.1016/j.na.2025.113756","DOIUrl":"10.1016/j.na.2025.113756","url":null,"abstract":"<div><div>We show the existence of the so-called semiclassical states <span><math><mrow><mi>U</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span> to the following curl–curl problem <span><math><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mo>∇</mo><mo>×</mo><mrow><mo>(</mo><mo>∇</mo><mo>×</mo><mi>U</mi><mo>)</mo></mrow><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>U</mi><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> for sufficiently small <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We study the asymptotic behaviour of solutions as <span><math><mrow><mi>ɛ</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span> and we investigate also a related nonlinear Schrödinger equation involving a singular potential. The problem models large permeability nonlinear materials satisfying the system of Maxwell equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113756"},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163864","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":"Radial positive solutions for mixed local and nonlocal supercritical Neumann problem","authors":"David Amundsen,&nbsp;Abbas Moameni,&nbsp;Remi Yvant Temgoua","doi":"10.1016/j.na.2025.113763","DOIUrl":"10.1016/j.na.2025.113763","url":null,"abstract":"<div><div>In this paper, we establish the existence of positive non-decreasing radial solutions for a nonlinear mixed local and nonlocal Neumann problem in the ball. No growth assumption on the nonlinearity is required. We also provide a criterion for the existence of non-constant solutions provided the problem possesses a trivial constant solution.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113763"},"PeriodicalIF":1.3,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163865","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":"Gradient Einstein-type warped products: Rigidity, existence and nonexistence results via a nonlinear PDE","authors":"José Nazareno Vieira Gomes ,&nbsp;Willian Isao Tokura","doi":"10.1016/j.na.2025.113759","DOIUrl":"10.1016/j.na.2025.113759","url":null,"abstract":"<div><div>We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry–Émery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113759"},"PeriodicalIF":1.3,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163860","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":"Asymptotics for positive singular solutions to subcritical sixth order equations","authors":"João Henrique Andrade ,&nbsp;Juncheng Wei","doi":"10.1016/j.na.2025.113757","DOIUrl":"10.1016/j.na.2025.113757","url":null,"abstract":"<div><div>We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. First, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden–Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables. The main difficulty is that the cylindrical PDE in this coordinate system is nonautonomous. Nonetheless, we define an associated nonautonomous Pohozaev functional, which can be proved to be asymptotically monotone. In addition, we show <em>a priori</em> estimates for these two functionals, from which we extract compactness properties. With these ingredients, we can perform an asymptotic analysis technique to prove our main result.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113757"},"PeriodicalIF":1.3,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163861","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":"Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian","authors":"Gyula Csató ,&nbsp;Albert Mas","doi":"10.1016/j.na.2025.113755","DOIUrl":"10.1016/j.na.2025.113755","url":null,"abstract":"<div><div>We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian <span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities <span><math><mi>f</mi></math></span> in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> and when <span><math><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mi>β</mi><mo>&lt;</mo><mn>1</mn></mrow></math></span>, the solution is not always <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>+</mo><mi>β</mi><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span> for all <span><math><mrow><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. Instead, in general the solution <span><math><mi>u</mi></math></span> is at most <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>β</mi><mo>)</mo></mrow></mrow></msup><mo>.</mo></mrow></math></span></div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"255 ","pages":"Article 113755"},"PeriodicalIF":1.3,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163862","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":"Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo’s law and non-convex flux","authors":"Junyuan Deng ,&nbsp;Lan Zhang","doi":"10.1016/j.na.2025.113749","DOIUrl":"10.1016/j.na.2025.113749","url":null,"abstract":"<div><div>This paper is concerned with the large time behavior of solutions to the scalar conservation law with an artificial heat flux term. The heat flux is governed by Cattaneo’s law, which leads to a 2 × 2 system of hyperbolic equations. The existence and nonlinear stability of rarefaction waves and viscous shock waves have been derived under the assumption that flux function is strictly convex. In the current paper, we focus on the one-dimensional Cauchy problem for the system which allows for non-convex flux. Under Oleinik entropy condition, we obtain the existence and asymptotic stability of shifted viscous shock waves with sufficiently small wave strength. The proof is based on the standard energy method and shift theory.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113749"},"PeriodicalIF":1.3,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136925","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":"Existence and boundedness of weak solutions to some vectorial Dirichlet problems","authors":"G.R. Cirmi ,&nbsp;S. D’Asero ,&nbsp;S. Leonardi ,&nbsp;F. Leonetti ,&nbsp;E. Rocha ,&nbsp;V. Staicu","doi":"10.1016/j.na.2025.113751","DOIUrl":"10.1016/j.na.2025.113751","url":null,"abstract":"<div><div>For integers <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mi>Ω</mi></math></span> a bounded subset of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, we prove existence and boundedness of a weak solution <span><math><mi>u</mi></math></span> of the following prototype of nonlinear vectorial Dirichlet problem <span><span><span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mspace></mspace><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>+</mo><msup><mrow><mi>F</mi></mrow><mrow><mi>ν</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>for any <span><math><mrow><mi>x</mi><mo>∈</mo><mi>Ω</mi></mrow></math></span> and <span><math><mrow><mi>ν</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></math></span>, where <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ν</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>ν</mi></mrow></msup></math></span> denote the <span><math><mrow><mi>ν</mi><mo>−</mo></mrow></math></span>th component of the vectors <span><math><mi>u</mi></math></span> and <span><math><mi>F</mi></math></span>, respectively, and the tensor <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math></span> satisfies suitable structural assumptions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113751"},"PeriodicalIF":1.3,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136923","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":"A regularity result for the Fokker–Planck equation with non-smooth drift and diffusion","authors":"Paolo Bonicatto ,&nbsp;Gennaro Ciampa ,&nbsp;Gianluca Crippa","doi":"10.1016/j.na.2025.113748","DOIUrl":"10.1016/j.na.2025.113748","url":null,"abstract":"<div><div>The goal of this paper is to study weak solutions of the Fokker–Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions’ theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of Figally (2008): on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker–Planck equation.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"254 ","pages":"Article 113748"},"PeriodicalIF":1.3,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143136924","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}