Nonlinear Analysis-Theory Methods & Applications最新文献

{"title":"Manifold-constrained free discontinuity problems and Sobolev approximation","authors":"Federico Luigi Dipasquale ,&nbsp;Bianca Stroffolini","doi":"10.1016/j.na.2024.113597","DOIUrl":"https://doi.org/10.1016/j.na.2024.113597","url":null,"abstract":"<div><p>We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano (Conti et al., 2017; Conti et al., 2019) for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001160/pdfft?md5=168447dad306b653853832564e192ce5&pid=1-s2.0-S0362546X24001160-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141540710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The trace fractional Laplacian and the mid-range fractional Laplacian","authors":"Julio D. Rossi ,&nbsp;Jorge Ruiz-Cases","doi":"10.1016/j.na.2024.113605","DOIUrl":"https://doi.org/10.1016/j.na.2024.113605","url":null,"abstract":"<div><p>In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem, that turns out to be nonlinear. Strong maximum and comparison principles also hold. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as <span><math><mrow><mi>s</mi><mo>↗</mo><mn>1</mn></mrow></math></span>, while for the second operator we obtain the sum of the smallest and the largest classical Hessian eigenvalues.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X2400124X/pdfft?md5=0652b7d18a04916d0cd30ffad56d2e9d&pid=1-s2.0-S0362546X2400124X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141540709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold","authors":"Jihye Lee ,&nbsp;Keomkyo Seo","doi":"10.1016/j.na.2024.113603","DOIUrl":"https://doi.org/10.1016/j.na.2024.113603","url":null,"abstract":"<div><p>We obtain Serrin-type theorems of the solution to overdetermined problems in a warped product manifold with a nonconstant Neumann boundary condition by applying the maximum principle to suitable subharmonic functions and integral identities.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141540708","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":"Well-posedness and inverse problems for semilinear nonlocal wave equations","authors":"Yi-Hsuan Lin ,&nbsp;Teemu Tyni ,&nbsp;Philipp Zimmermann","doi":"10.1016/j.na.2024.113601","DOIUrl":"https://doi.org/10.1016/j.na.2024.113601","url":null,"abstract":"<div><p>This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001202/pdfft?md5=514b86014fbe067975460c2eb5bc96b4&pid=1-s2.0-S0362546X24001202-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"L∞ blow-up in the Jordan–Moore–Gibson–Thompson equation","authors":"Vanja Nikolić ,&nbsp;Michael Winkler","doi":"10.1016/j.na.2024.113600","DOIUrl":"https://doi.org/10.1016/j.na.2024.113600","url":null,"abstract":"<div><p>The Jordan–Moore–Gibson–Thompson equation <span><span><span><math><mrow><mi>τ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>β</mi><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>γ</mi><mi>Δ</mi><mi>u</mi><mo>+</mo><msub><mrow><mrow><mo>(</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub></mrow></math></span></span></span>is considered in a smoothly bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> with <span><math><mrow><mi>n</mi><mo>≤</mo><mn>3</mn></mrow></math></span>, where <span><math><mrow><mi>τ</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>β</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo><mi>γ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, and <span><math><mrow><mi>α</mi><mo>∈</mo><mi>R</mi></mrow></math></span>.</p><p>Firstly, it is seen that under the assumption that <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> is such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial–boundary value problem admits a unique solution <span><math><mi>u</mi></math></span> on a maximal time interval <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow></math></span> which is such that <span><span><span><math><mrow><mtext>if</mtext><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>&lt;</mo><mi>∞</mi><mo>,</mo><mspace></mspace><mtext>then</mtext><mspace></mspace><munder><mrow><mo>lim sup</mo></mrow><mrow><mi>t</mi><mo>↗</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></munder><msub><mrow><mo>‖</mo><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo>=</mo><mi>∞</mi><mo>.</mo><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mo>⋆</mo><mo>)</mo></mrow></mrow></math></span></span></span>This is used to, secondly, make sure that if additionally <span><math><mi>f</mi></math></span> is convex and grows superlinearly in the sense that <span><span><span><math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>≥</mo><mn>0</mn><mtext>on</mtext><mi>R</mi><mtext>,</mtext><mspace></mspace><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mi>ξ</","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001196/pdfft?md5=958a8da5d648564aa56ba997c9422290&pid=1-s2.0-S0362546X24001196-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics","authors":"Suzhen Mao ,&nbsp;Aliang Xia ,&nbsp;Yan Xu","doi":"10.1016/j.na.2024.113602","DOIUrl":"https://doi.org/10.1016/j.na.2024.113602","url":null,"abstract":"<div><p>In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: <span><span><span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mfrac><mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><msub><mrow><mn>2</mn></mrow><mrow><mo>∗</mo></mrow></msub><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow></mrow></mfrac><mo>,</mo><mspace></mspace><mi>x</mi><mo>=</mo><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>n</mi><mo>&gt;</mo><mn>4</mn></mrow></math></span>, <span><math><mrow><mn>2</mn><mo>≤</mo><mi>m</mi><mo>&lt;</mo><mi>n</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mn>2</mn></mrow><mrow><mo>∗</mo></mrow></msub><mo>≔</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span> and potential function <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>R</mi></mrow></math></span>. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485471","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":"On the decaying property of quintic NLS on 3D hyperbolic space","authors":"Chutian Ma ,&nbsp;Han Wang ,&nbsp;Xueying Yu ,&nbsp;Zehua Zhao","doi":"10.1016/j.na.2024.113599","DOIUrl":"https://doi.org/10.1016/j.na.2024.113599","url":null,"abstract":"<div><p>In this paper, we study the (pointwise) decaying property of quintic NLS on the three-dimensional hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. We show the nonlinear solution enjoys the same decay rate as the linear solution does. This result is based on the associated global well-posedness and scattering result in Ionescu et al. (2012). This extends (Fan and Zhao, 2021)’ Euclidean works to the hyperbolic space with additional improvements in regularity requirement (lower and almost critical regularity assumed). Realizing such improvements also work for the Euclidean case, we obtain a result for the fourth-order NLS analogue studied in Yu et al. (2023) recently with better, i.e. almost critical regularity assumption.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485470","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":"Long-time dynamics for the energy critical heat equation in R5","authors":"Zaizheng Li ,&nbsp;Juncheng Wei ,&nbsp;Qidi Zhang ,&nbsp;Yifu Zhou","doi":"10.1016/j.na.2024.113594","DOIUrl":"https://doi.org/10.1016/j.na.2024.113594","url":null,"abstract":"<div><p>We investigate the long-time behavior of global solutions to the energy critical heat equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><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><mspace></mspace><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>×</mo><mrow><mo>(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mspace></mspace><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>For <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> sufficiently large, we show the existence of positive solutions for a class of initial value <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∼</mo><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup></mrow></math></span> as <span><math><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>→</mo><mi>∞</mi></mrow></math></span> with <span><math><mrow><mi>γ</mi><mo>&gt;</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> such that the global solutions behave asymptotically <span><span><span><math><mrow><msub><mrow><mo>‖</mo><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>)</mo></mrow></mrow></msub><mo>∼</mo><mfenced><mrow><mtable><mtr><mtd><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>γ</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mspace></mspace><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mi>γ</mi><mo>&lt;</mo><mn>2</mn></mtd></mtr><mtr><mtd><msup><mrow><mrow><mo>(</mo><mo>ln</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mspace></mspace><mspace></mspace></mtd><mtd><mtext>if</mtext><mspace></mspace><mi>γ</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn><mspace></mspace><mspace></mspace></mtd><mtd><mtext>if</mtext><msp","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485469","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":"Exploring a modification of dp convergence","authors":"Brian Allen ,&nbsp;Edward Bryden","doi":"10.1016/j.na.2024.113598","DOIUrl":"https://doi.org/10.1016/j.na.2024.113598","url":null,"abstract":"<div><p>In the work by M. C. Lee, A. Naber, and R. Neumayer (Lee et al., 2023) a beautiful <span><math><mi>ɛ</mi></math></span>-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> distance for Riemannian manifolds is introduced and the quantitative stability results are given in terms of this notion of distance, with important examples showing why other existing notions of convergence are not adequate in their setting. Due to the presence of an entropy bound, the possibility of long, thin splines forming along a sequence of Riemannian manifolds whose scalar curvature is becoming almost positive is ruled out. It is important to rule out such examples since the <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> distance is not well behaved in the presences of splines that persist in the limit. Since there are many geometric stability conjectures (Gromov, 2023; Sormani, 2023) where we want to allow for the presence of splines that persist in the limit, it is crucial to be able to modify the <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> distance to retain its positive qualities and prevent it from being sensitive to splines. In this paper we explore one such modification of the <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> distance and give a theorem which allows one to estimate the modified <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> distance, which we expect to be useful in practice.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485061","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 note on Kazdan–Warner type equations on compact Riemannian manifolds","authors":"Weike Yu","doi":"10.1016/j.na.2024.113596","DOIUrl":"https://doi.org/10.1016/j.na.2024.113596","url":null,"abstract":"<div><p>In this note, we prove an existence result for generalized Kazdan–Warner equations on compact Riemannian manifolds by using the flow approach or the upper and lower solution method. In addition, we give a priori estimates for this type equations.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141424309","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}