Advanced Nonlinear Studies最新文献

{"title":"Decay estimates for defocusing energy-critical Hartree equation","authors":"Miao Chen, Hua Wang, Xiaohua Yao","doi":"10.1515/ans-2023-0138","DOIUrl":"https://doi.org/10.1515/ans-2023-0138","url":null,"abstract":"In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>${H}^{2}left({mathbb{R}}^{5}right)cap {L}^{1}left({mathbb{R}}^{5}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0138_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula>. We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"21 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840659","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 of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials","authors":"Caifeng Zhang, Maochun Zhu","doi":"10.1515/ans-2023-0136","DOIUrl":"https://doi.org/10.1515/ans-2023-0136","url":null,"abstract":"The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint:&lt;jats:disp-formula&gt; &lt;jats:label&gt;(0.1)&lt;/jats:label&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mi&gt;sup&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∫&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;∇&lt;/m:mi&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;d&lt;/m:mi&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∫&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;e&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;4&lt;/m:mn&gt; &lt;m:mi&gt;π&lt;/m:mi&gt; &lt;m:mfenced close=\")\" open=\"(\"&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:mi mathvariant=\"normal\"&gt;d&lt;/m:mi&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;∞&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt; $$underset{uin {H}^{1}({mathbb{R}}^{2}),underset{{mathbb{R}}^{2}}{int }(vert nabla u{vert }^{2}+V(x){u}^{2})mathrm{d}xle 1}{mathrm{sup}}underset{{mathbb{R}}^{2}}{int }frac{{e}^{4pi left(1-tfrac{beta }{2}right){u}^{2}}-1}{vert x{vert }^{beta }}mathrm{d}x&lt; +infty ,$$ &lt;/jats:tex-math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0136_eq_001.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:mn&gt;0&lt;/m:m","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"16 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840657","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 degenerate migration-consumption model in domains of arbitrary dimension","authors":"Michael Winkler","doi":"10.1515/ans-2023-0131","DOIUrl":"https://doi.org/10.1515/ans-2023-0131","url":null,"abstract":"In a smoothly bounded convex domain &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:mi mathvariant=\"normal\"&gt;Ω&lt;/m:mi&gt; &lt;m:mo&gt;⊂&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;${Omega}subset {mathbb{R}}^{n}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_ineq_001.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; with &lt;jats:italic&gt;n&lt;/jats:italic&gt; ≥ 1, a no-flux initial-boundary value problem for&lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"&gt; &lt;m:mfenced close=\"\" open=\"{\"&gt; &lt;m:mrow&gt; &lt;m:mtable&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mfenced close=\")\" open=\"(\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mi&gt;ϕ&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"1em\"/&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"1em\"/&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;/m:mtable&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;$$begin{cases}_{t}={Delta}left(uphi left(vright)right),quad hfill {v}_{t}={Delta}v-uv,quad hfill end{cases}$$&lt;/jats:tex-math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_eq_999.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;is considered under the assumption that near the origin, the function &lt;jats:italic&gt;ϕ&lt;/jats:italic&gt; suitably generalizes the prototype given by&lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"&gt; &lt;m:mi&gt;ϕ&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"2em\"/&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;[&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;]&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;.&lt;/m:mo&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;$$phi left(xi right)={xi }^{alpha },qquad xi in left[0,{xi }_{0}right].$$&lt;/jats:tex-math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_eq_998.png\"/&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt;By means of separate approaches, it is shown that in bo","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"64 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840862","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":"New multiplicity results in prescribing Q-curvature on standard spheres","authors":"Mohamed Ben Ayed, Khalil El Mehdi","doi":"10.1515/ans-2023-0135","DOIUrl":"https://doi.org/10.1515/ans-2023-0135","url":null,"abstract":"In this paper, we study the problem of prescribing <jats:italic>Q</jats:italic>-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function <jats:italic>K</jats:italic> so that it is the <jats:italic>Q</jats:italic>-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function <jats:italic>K</jats:italic>.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"51 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623905","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":"Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system","authors":"Francisco Ortegón Gallego, Mohamed Rhoudaf, Hajar Talbi","doi":"10.1515/ans-2023-0133","DOIUrl":"https://doi.org/10.1515/ans-2023-0133","url":null,"abstract":"In this paper, we analyze the following nonlinear elliptic problem <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mfenced close=\"\" open=\"{\"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>∇</m:mi> <m:mi>φ</m:mi> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mtext> in </m:mtext> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mtext>div</m:mtext> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>∇</m:mi> <m:mi>φ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> in </m:mtext> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>φ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>.</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>$begin{cases}Aleft(uright)=rho left(uright)vert nabla varphi {vert }^{2},text{in},{Omega},quad hfill text{div}left(rho left(uright)nabla varphi right)=0,text{in},{Omega},quad hfill u=0,text{on},partial {Omega},quad hfill varphi ={varphi }_{0},text{on},partial {Omega}.quad hfill end{cases}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0133_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> where <jats:italic>A</jats:italic>(<jats:italic>u</jats:italic>) = −div <jats:italic>a</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>u</jats:italic>, ∇<jats:italic>u</jats:italic>) is a Leray-Lions operator of order <jats:italic>p</jats:italic>. The second member of the first equation is only in <jats:italic>L</jats:italic> <jats:sup>1</jats:sup>(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"9 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140624049","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":"Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces","authors":"Ali Taheri, Vahideh Vahidifar","doi":"10.1515/ans-2023-0120","DOIUrl":"https://doi.org/10.1515/ans-2023-0120","url":null,"abstract":"In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"2 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579549","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":"Multiple concentrating solutions for a fractional (p, q)-Choquard equation","authors":"Vincenzo Ambrosio","doi":"10.1515/ans-2023-0125","DOIUrl":"https://doi.org/10.1515/ans-2023-0125","url":null,"abstract":"We focus on the following fractional (&lt;jats:italic&gt;p&lt;/jats:italic&gt;, &lt;jats:italic&gt;q&lt;/jats:italic&gt;)-Choquard problem: &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mfenced open=\"{\" close=\"\"&gt; &lt;m:mrow&gt; &lt;m:mtable&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mfenced open=\"(\" close=\")\"&gt; &lt;m:mrow&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;μ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;m:mi&gt;F&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mspace width=\"0.17em\" /&gt; &lt;m:mtext&gt; in &lt;/m:mtext&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"1em\" /&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;W&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∩&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;W&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"89 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579814","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 large solutions to a class of k-Hessian problems","authors":"Haitao Wan","doi":"10.1515/ans-2023-0128","DOIUrl":"https://doi.org/10.1515/ans-2023-0128","url":null,"abstract":"In this paper, we consider the &lt;jats:italic&gt;k&lt;/jats:italic&gt;-Hessian problem &lt;jats:italic&gt;S&lt;/jats:italic&gt; &lt;jats:sub&gt; &lt;jats:italic&gt;k&lt;/jats:italic&gt; &lt;/jats:sub&gt;(&lt;jats:italic&gt;D&lt;/jats:italic&gt; &lt;jats:sup&gt;2&lt;/jats:sup&gt; &lt;jats:italic&gt;u&lt;/jats:italic&gt;) = &lt;jats:italic&gt;b&lt;/jats:italic&gt;(&lt;jats:italic&gt;x&lt;/jats:italic&gt;)&lt;jats:italic&gt;f&lt;/jats:italic&gt;(&lt;jats:italic&gt;u&lt;/jats:italic&gt;) in Ω, &lt;jats:italic&gt;u&lt;/jats:italic&gt; = +∞ on &lt;jats:italic&gt;∂&lt;/jats:italic&gt;Ω, where Ω is a &lt;jats:italic&gt;C&lt;/jats:italic&gt; &lt;jats:sup&gt;∞&lt;/jats:sup&gt;-smooth bounded strictly (&lt;jats:italic&gt;k&lt;/jats:italic&gt; − 1)-convex domain in &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;${mathbb{R}}^{N}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0128_ineq_001.png\" /&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; with &lt;jats:italic&gt;N&lt;/jats:italic&gt; ≥ 2, &lt;jats:italic&gt;b&lt;/jats:italic&gt; ∈ C&lt;jats:sup&gt;∞&lt;/jats:sup&gt;(Ω) is positive in Ω and may be singular or vanish on &lt;jats:italic&gt;∂&lt;/jats:italic&gt;Ω, &lt;jats:italic&gt;f&lt;/jats:italic&gt; ∈ &lt;jats:italic&gt;C&lt;/jats:italic&gt;[0, ∞) ∩ &lt;jats:italic&gt;C&lt;/jats:italic&gt; &lt;jats:sup&gt;1&lt;/jats:sup&gt;(0, ∞) (or &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;C&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;$fin {C}^{1}left(mathbb{R}right)$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0128_ineq_002.png\" /&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;) is a positive and increasing function. We establish the first expansions (equalities) of &lt;jats:italic&gt;k&lt;/jats:italic&gt;-convex solutions to the above problem when &lt;jats:italic&gt;f&lt;/jats:italic&gt; is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of &lt;jats:italic&gt;f&lt;/jats:italic&gt; and principal curvatures of &lt;jats:italic&gt;∂&lt;/jats:italic&gt;Ω on the first expansion of solutions. For the latter, we find the principal curvatures of &lt;jats:italic&gt;∂&lt;/jats:italic&gt;Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including &lt;jats:italic&gt;k&lt;/jats:italic&gt; = &lt;jats:italic&gt;N&lt;/jats:italic&gt;). Moreover, we know the existence of &lt;jats:italic&gt;k&lt;/jats:italic&gt;-convex solutions to the above problem (including &lt;jats:italic&gt;k&lt;/jats:italic&gt; = &lt;jats:italic&gt;N&lt;/jats:italic&gt;) is still an open problem when &lt;jats:italic&gt;b&lt;/jats:italic&gt; possesses high singularity on &lt;jats:italic&gt;∂&lt;/jats:italic&gt;Ω and &lt;jats:italic&gt;f&lt;/jats:italic&gt; satisfies Keller–Os","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"49 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579435","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":"Michael-Simon type inequalities in hyperbolic space H n + 1 ${mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows","authors":"Jingshi Cui, Peibiao Zhao","doi":"10.1515/ans-2023-0127","DOIUrl":"https://doi.org/10.1515/ans-2023-0127","url":null,"abstract":"In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;${mathbb{H}}^{n+1}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0127_ineq_002.png\" /&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;H&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;${mathbb{H}}^{n+1}$&lt;/jats:tex-math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0127_ineq_003.png\" /&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;,” (Preprint)) as follows&lt;jats:disp-formula&gt; &lt;jats:label&gt;(0.1)&lt;/jats:label&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∫&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;′&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:msqrt&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;E&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;∇&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:msqrt&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∫&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:mfenced close=\"⟩\" open=\"⟨\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mover accent=\"true\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;∇&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;̄&lt;/m:mo&gt; &lt;/m:mover&gt; &lt;/m:mrow&gt; &lt;m:mfenced close=\")\" open=\"(\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;′&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;ν&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:munder&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∫&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;∂&lt;/m:mi&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;≥&lt;/m:mo&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ω&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mfenced close=\")\" open=\"(\"&gt; &lt;m:mrow","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"32 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579431","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":"Moving planes and sliding methods for fractional elliptic and parabolic equations","authors":"Wenxiong Chen, Yeyao Hu, Lingwei Ma","doi":"10.1515/ans-2022-0069","DOIUrl":"https://doi.org/10.1515/ans-2022-0069","url":null,"abstract":"In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168008","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}