{"title":"Multi-bump solutions for the magnetic Schrödinger–Poisson system with critical growth","authors":"Chao Ji, YongDe Zhang, V. Rǎdulescu","doi":"10.14232/ejqtde.2022.1.21","DOIUrl":null,"url":null,"abstract":"<jats:p>In this paper, we are concerned with the following magnetic Schrödinger–Poisson system <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>u</mml:mi> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mtext> in </mml:mtext> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mtext> in </mml:mtext> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> is a parameter, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>f</mml:mi> </mml:math> is a subcritical nonlinearity, the potential <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:math> is a continuous function verifying some conditions, the magnetic potential <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>A</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> . Assuming that the zero set of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> has several isolated connected components <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> </mml:math> such that the interior of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:math> is non-empty and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:msub> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:math> is smooth, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>j</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> </mml:math>, then for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> large enough, we use the variational methods to show that the above system has at least <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:math> multi-bump solutions.</jats:p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2022.1.21","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we are concerned with the following magnetic Schrödinger–Poisson system {−(∇+iA(x))2u+(λV(x)+1)u+ϕu=αf(|u|2)u+|u|4u, in R3,−Δϕ=u2, in R3, where λ>0 is a parameter, f is a subcritical nonlinearity, the potential V:R3→R is a continuous function verifying some conditions, the magnetic potential A∈Lloc2(R3,R3) . Assuming that the zero set of V(x) has several isolated connected components Ω1,…,Ωk such that the interior of Ωj is non-empty and ∂Ωj is smooth, where j∈{1,…,k}, then for λ>0 large enough, we use the variational methods to show that the above system has at least 2k−1 multi-bump solutions.
在本文中，我们关注以下磁性Schrödinger-Poisson系统{−(∇+ i A (x)) 2u + (λ V)(x) + 1) u + φ u = α f (| u | 2) u + | u在r3中，−Δ ϕ = u 2，在R 3中，其中λ > 0为参数，f为亚临界非线性，势V: r3→R为连续函数，验证某些条件，磁势a∈L L o c2 (r3, r3)。假设V (x)的零集有几个孤立的连通分量Ω 1，…，Ω k，使得Ω j的内部是非空的，∂Ω j是光滑的，其中j∈{1，…，k}，那么对于λ >足够大，我们使用变分方法证明了上述系统至少有2k−1个多凹凸解。
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