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Self-similar solutions for the generalized fractional Korteweg–de Vries equation 广义分数阶Korteweg-de Vries方程的自相似解
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-31 DOI: 10.1016/j.na.2025.113906
Luc Molinet , Stéphane Vento , Fred Weissler
{"title":"Self-similar solutions for the generalized fractional Korteweg–de Vries equation","authors":"Luc Molinet ,&nbsp;Stéphane Vento ,&nbsp;Fred Weissler","doi":"10.1016/j.na.2025.113906","DOIUrl":"10.1016/j.na.2025.113906","url":null,"abstract":"<div><div>We consider the Cauchy problem for the generalized fractional Korteweg–de Vries equation <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>&lt;</mo><mi>α</mi><mo>≤</mo><mn>2</mn><mo>,</mo><mspace></mspace><mi>p</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> with homogeneous initial data <span><math><mi>Φ</mi></math></span>. We show that, under smallness assumption on <span><math><mi>Φ</mi></math></span>, and for a wide range of <span><math><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></math></span>, including <span><math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math></span>, we can construct a self-similar solution of this problem.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113906"},"PeriodicalIF":1.3,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748769","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}
引用次数: 0
One-dimensional half-harmonic maps into the circle and their degree 一维半谐波映射到圆和它们的度
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-30 DOI: 10.1016/j.na.2025.113904
Ali Hyder , Luca Martinazzi
{"title":"One-dimensional half-harmonic maps into the circle and their degree","authors":"Ali Hyder ,&nbsp;Luca Martinazzi","doi":"10.1016/j.na.2025.113904","DOIUrl":"10.1016/j.na.2025.113904","url":null,"abstract":"<div><div>Given a half-harmonic map <span><math><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in <span><math><mrow><mi>R</mi><mo>∖</mo><mi>I</mi></mrow></math></span>, we show the existence of a second half-harmonic map, minimizing the fractional Dirichlet energy in a different homotopy class. This is based on the study of the degree of fractional Sobolev maps and a sharp estimate à la Brezis-Coron. We give examples showing that it is in general not possible to minimize in every homotopy class and show a contrast with the 2-dimensional case.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113904"},"PeriodicalIF":1.3,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144723157","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}
引用次数: 0
Elliptic Harnack inequality and its applications on Finsler metric measure spaces 椭圆型哈纳克不等式及其在Finsler度量测度空间上的应用
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-26 DOI: 10.1016/j.na.2025.113907
Xinyue Cheng, Liulin Liu, Yu Zhang
{"title":"Elliptic Harnack inequality and its applications on Finsler metric measure spaces","authors":"Xinyue Cheng,&nbsp;Liulin Liu,&nbsp;Yu Zhang","doi":"10.1016/j.na.2025.113907","DOIUrl":"10.1016/j.na.2025.113907","url":null,"abstract":"<div><div>In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature <span><math><msub><mrow><mi>Ric</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> has non-positive lower bound and the distortion <span><math><mi>τ</mi></math></span> is of linear growth, <span><math><mrow><mrow><mo>|</mo><mi>τ</mi><mo>|</mo></mrow><mo>≤</mo><mi>a</mi><mi>r</mi><mo>+</mo><mi>b</mi></mrow></math></span>, where <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></math></span> are some non-negative constants, <span><math><mrow><mi>r</mi><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is the distance function for some point <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>M</mi></mrow></math></span>. We obtain an elliptic <span><math><mi>p</mi></math></span>-Harnack inequality for positive harmonic functions from a local uniform Poincaré inequality and a mean value inequality. As applications of the Harnack inequality, we derive the Hölder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113907"},"PeriodicalIF":1.3,"publicationDate":"2025-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713310","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}
引用次数: 0
Hodge theory on the Harmonic Gasket and other fractals 调和垫片和其他分形的霍奇理论
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-25 DOI: 10.1016/j.na.2025.113892
Ugo Bessi
{"title":"Hodge theory on the Harmonic Gasket and other fractals","authors":"Ugo Bessi","doi":"10.1016/j.na.2025.113892","DOIUrl":"10.1016/j.na.2025.113892","url":null,"abstract":"<div><div>S. Kusuoka has proven that, on many fractals <span><math><mrow><mi>G</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, it is possible to build a natural bilinear form on the vector space of Borel fields of one-forms on <span><math><mi>G</mi></math></span>. A variant of this construction yields a bilinear form on Borel fields of <span><math><mi>q</mi></math></span>-forms; it is tempting to ask (and several authors have done it) whether some features of Hodge theory survive in this setting. In this paper we define a weak version of the codifferential on fractals and we show that, for one-forms on the Harmonic Sierpinski Gasket, a Hodge decomposition theorem holds. As a further example, we calculate the codifferential of 2-forms and 1-forms on a fractal of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> which is the product of the harmonic Sierpinski gasket with the interval <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113892"},"PeriodicalIF":1.3,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702222","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}
引用次数: 0
Diffusion on d-sphere: Periodicity and inertial manifolds d球上的扩散：周期性和惯性流形
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-22 DOI: 10.1016/j.na.2025.113889
Thieu Huy Nguyen, Thi Ngoc Ha Vu
{"title":"Diffusion on d-sphere: Periodicity and inertial manifolds","authors":"Thieu Huy Nguyen,&nbsp;Thi Ngoc Ha Vu","doi":"10.1016/j.na.2025.113889","DOIUrl":"10.1016/j.na.2025.113889","url":null,"abstract":"<div><div>On a <span><math><mi>d</mi></math></span>-sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we consider the diffusion equation of the form <span><math><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>H</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mi>♭</mi></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>♯</mi></mrow></msup><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> for vector fields on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and Hodge Laplacian <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span>. We prove the existence of a <span><math><mi>T</mi></math></span>-periodic solution to that equation under the action of a <span><math><mi>T</mi></math></span>-periodic external force <span><math><mi>g</mi></math></span>. Furthermore, we investigate the existence of an inertial manifold for the solutions nearby that periodic solution. The distribution of eigenvalues of Hodge Laplacian leads to the validity of the spectral gap condition yielding the existence of an inertial manifold.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113889"},"PeriodicalIF":1.3,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679141","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}
引用次数: 0
Further applications of the Nehari manifold method to functionals in C1(X∖{0}) Nehari流形法在C1（X ×{0}）泛函中的进一步应用
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-22 DOI: 10.1016/j.na.2025.113890
Edir Júnior Ferreira Leite , Humberto Ramos Quoirin , Kaye Silva
{"title":"Further applications of the Nehari manifold method to functionals in C1(X∖{0})","authors":"Edir Júnior Ferreira Leite ,&nbsp;Humberto Ramos Quoirin ,&nbsp;Kaye Silva","doi":"10.1016/j.na.2025.113890","DOIUrl":"10.1016/j.na.2025.113890","url":null,"abstract":"<div><div>We proceed with the study of the Nehari manifold method for functionals in <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>X</mi></math></span> is a Banach space. We deal now with functionals whose fibering maps have two critical points (a minimizer followed by a maximizer). Under some additional conditions we show that the Nehari manifold method provides us with the ground state level and two sequences of critical values for these functionals. These results are applied to the class of <em>prescribed energy problems</em> as well as to the concave–convex problem for the <em>affine</em> <span><math><mi>p</mi></math></span>-Laplacian operator.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113890"},"PeriodicalIF":1.3,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679142","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}
引用次数: 0
Note on Fourier inequalities 关于傅里叶不等式的注意事项
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-19 DOI: 10.1016/j.na.2025.113896
Miquel Saucedo , Sergey Tikhonov
{"title":"Note on Fourier inequalities","authors":"Miquel Saucedo ,&nbsp;Sergey Tikhonov","doi":"10.1016/j.na.2025.113896","DOIUrl":"10.1016/j.na.2025.113896","url":null,"abstract":"<div><div>We prove that the Hausdorff–Young inequality <span><math><mrow><msub><mrow><mo>‖</mo><mover><mrow><mi>f</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>‖</mo></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow></msub><mo>≤</mo><mi>C</mi><msub><mrow><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow></msub></mrow></math></span> with <span><math><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>⋅</mi><mo>)</mo></mrow></mrow></math></span> even and non-decreasing holds in variable Lebesgue spaces if and only if <span><math><mi>p</mi></math></span> is a constant. However, under the additional condition on monotonicity of <span><math><mi>f</mi></math></span>, we obtain a complete characterization of Pitt-type weighted Fourier inequalities in both the classical and variable Lebesgue setting.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113896"},"PeriodicalIF":1.3,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144665925","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}
引用次数: 0
A family of explicit minimizers for interaction energies 相互作用能的显式极小值族
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-16 DOI: 10.1016/j.na.2025.113900
Ruiwen Shu
{"title":"A family of explicit minimizers for interaction energies","authors":"Ruiwen Shu","doi":"10.1016/j.na.2025.113900","DOIUrl":"10.1016/j.na.2025.113900","url":null,"abstract":"<div><div>In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials <span><math><mrow><mi>W</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>a</mi></mrow></msup></mrow><mrow><mi>a</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>b</mi></mrow></msup></mrow><mrow><mi>b</mi></mrow></mfrac></mrow></math></span> in <span><math><mi>d</mi></math></span> dimensions. For odd <span><math><mi>d</mi></math></span> with <span><math><mrow><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>−</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span> and even <span><math><mi>d</mi></math></span> with <span><math><mrow><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>−</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>, we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler–Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions <span><math><mi>d</mi></math></span>, the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension <span><math><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></math></span> via a new lemma on dimension reduction.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113900"},"PeriodicalIF":1.3,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633185","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}
引用次数: 0
Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term 带强迫项的各向异性平均曲率流动的阈值动力学近似格式
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-16 DOI: 10.1016/j.na.2025.113899
Bohdan Bulanyi, Berardo Ruffini
{"title":"Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term","authors":"Bohdan Bulanyi,&nbsp;Berardo Ruffini","doi":"10.1016/j.na.2025.113899","DOIUrl":"10.1016/j.na.2025.113899","url":null,"abstract":"<div><div>We establish the convergence of threshold dynamics-type approximation schemes to propagating fronts evolving according to an anisotropic mean curvature motion in the presence of a forcing term depending on both time and position, thus generalizing the consistency result obtained in Ishii, Pires and Souganidis (1999) by extending the results obtained in Caffarelli and Souganidis (2010) for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> to anisotropic kernels and in the presence of a driving force. The limit geometric evolution is of a variational type and can be approximated, at a large scale, by eikonal-type equations modeling dislocations dynamics. We prove that it preserves convexity under suitable convexity assumptions on the forcing term and that convex evolutions of compact sets are unique. If the initial set is bounded and sufficiently large, and the driving force is constant, then the corresponding generalized front propagation is asymptotically similar to the Wulff shape.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113899"},"PeriodicalIF":1.3,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633186","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}
引用次数: 0
A note on entire large solutions to semilinear elliptic systems of competitive type 半线性竞争型椭圆系统的整体大解的注释
IF 1.3 2区 数学
Nonlinear Analysis-Theory Methods & Applications Pub Date : 2025-07-15 DOI: 10.1016/j.na.2025.113903
Alan V. Lair
{"title":"A note on entire large solutions to semilinear elliptic systems of competitive type","authors":"Alan V. Lair","doi":"10.1016/j.na.2025.113903","DOIUrl":"10.1016/j.na.2025.113903","url":null,"abstract":"<div><div>We consider the elliptic system <span><math><mrow><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>p</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi>v</mi></mrow><mrow><mi>b</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>Δ</mi><mi>v</mi><mo>=</mo><mi>q</mi><mrow><mo>(</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>c</mi></mrow></msup><msup><mrow><mi>v</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span> on <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>3</mn></mrow></math></span>) where <span><math><mi>a</mi></math></span>, <span><math><mi>b</mi></math></span>, <span><math><mi>c</mi></math></span>, <span><math><mi>d</mi></math></span> are positive constants with <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>}</mo></mrow><mo>&gt;</mo><mn>1</mn></mrow></math></span>, and the functions <span><math><mi>p</mi></math></span> and <span><math><mi>q</mi></math></span> are positive and continuous. We establish conditions on <span><math><mi>p</mi></math></span> and <span><math><mi>q</mi></math></span>, along with the exponents <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></math></span>, which ensure the existence of a positive entire solution satisfying <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>→</mo><mi>∞</mi></mrow></msub><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>→</mo><mi>∞</mi></mrow></msub><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>∞</mi></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113903"},"PeriodicalIF":1.3,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631329","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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