{"title":"A non-trivial solution for a p-Schrödinger–Kirchhoff-type integro-differential system by non-smooth techniques","authors":"Juan Mayorga-Zambrano, Daniel Narváez-Vaca","doi":"10.1007/s43034-023-00299-5","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the integro-differential system <span>\\((\\textrm{P}_m)\\)</span>: </p><div><div><span>$$\\begin{aligned} - \\left( a_k+b_k \\left( \\displaystyle \\int _{{\\mathbb {R}}^{N}} |\\nabla u_k|^{p} dx \\right) ^{p-1} \\right) \\Delta _{p} u_k + V(x) |u_k|^{p-2} u_k = \\partial _{k} F(u_1,\\ldots ,u_m), \\end{aligned}$$</span></div></div><p>where <span>\\(x\\in {\\mathbb {R}}^N\\)</span>, <span>\\(a_k>0\\)</span>, <span>\\(b_k\\ge 0\\)</span>, <span>\\(N\\ge 2\\)</span> and <span>\\(1<p<N\\)</span>, <span>\\(u_k \\in \\textrm{W}^{1,p}({\\mathbb {R}}^{N})\\)</span>, for <span>\\(k=1,\\ldots ,m\\)</span>. By <span>\\(\\partial _{k} F(u_1,\\ldots ,u_m),\\)</span> it is denoted the <i>k</i>-th partial generalized gradient in the sense of Clarke. The potential <span>\\(V\\in \\textrm{C} \\left( {\\mathbb {R}}^N \\right) \\)</span> verifies <span>\\(\\inf (V)>0\\)</span> and a coercivity property introduced by Bartsch et al. The coupling function <span>\\(F:{\\mathbb {R}}^m\\longrightarrow {\\mathbb {R}}\\)</span> is locally Lipschitz and verifies conditions introduced by Duan and Huang. By applying tools from the non-smooth critical point theory, we prove the existence of a non-trivial mountain pass solution of <span>\\((\\textrm{P}_m)\\)</span>.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00299-5","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the integro-differential system \((\textrm{P}_m)\):
where \(x\in {\mathbb {R}}^N\), \(a_k>0\), \(b_k\ge 0\), \(N\ge 2\) and \(1<p<N\), \(u_k \in \textrm{W}^{1,p}({\mathbb {R}}^{N})\), for \(k=1,\ldots ,m\). By \(\partial _{k} F(u_1,\ldots ,u_m),\) it is denoted the k-th partial generalized gradient in the sense of Clarke. The potential \(V\in \textrm{C} \left( {\mathbb {R}}^N \right) \) verifies \(\inf (V)>0\) and a coercivity property introduced by Bartsch et al. The coupling function \(F:{\mathbb {R}}^m\longrightarrow {\mathbb {R}}\) is locally Lipschitz and verifies conditions introduced by Duan and Huang. By applying tools from the non-smooth critical point theory, we prove the existence of a non-trivial mountain pass solution of \((\textrm{P}_m)\).
期刊介绍:
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
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