Scholarlink

<p>In this article, we study the multiple solutions of a class of variable-order Schrödinger–Kirchhoff-type double-phase system, the equation has a nonlinear term of the concave–convex nonlinearities with variable exponent and a new type critical term which is better suitable for double-phase problem. Using the concentration-compactness principle and Kajikiya’s symmetric mountain pass theorem, the existence of infinitely many solutions for suitable small parameters <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_438_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_438_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nu _i,i=1,2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ν</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> has been obtained, respectively. This implies that infinite solutions exist when the parameters <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_438_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mu\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_438_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">\(\max \{ \nu _1,\nu _2\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mn>2</mn> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> lie within an <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_438_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">L</mi> </math></EquationSource> </InlineEquation>-shaped region (see Fig.&#xa0;<InternalRef RefID="Fig1">1</InternalRef>). A technique is developed to determine the geometry of energy functionals in such Schrödinger–Kirchhoff-type systems with concave–convex terms and variable exponents.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Infinitely solutions for a variable-order Schrödinger–Kirchhoff-type double-phase system with new critical growth in \(\mathbb {R}^n\)

  • 期刊论文

  • 发表日期：

    2025年7月11日

查看全文
  • Yizhe Feng,
  • Zhanbing Bai

摘要

In this article, we study the multiple solutions of a class of variable-order Schrödinger–Kirchhoff-type double-phase system, the equation has a nonlinear term of the concave–convex nonlinearities with variable exponent and a new type critical term which is better suitable for double-phase problem. Using the concentration-compactness principle and Kajikiya’s symmetric mountain pass theorem, the existence of infinitely many solutions for suitable small parameters \(\mu\) μ and \(\nu _i,i=1,2\) ν i , i = 1 , 2 has been obtained, respectively. This implies that infinite solutions exist when the parameters \(\mu\) μ and \(\max \{ \nu _1,\nu _2\}\) max { ν 1 , ν 2 } lie within an \(\mathbb {L}\) L -shaped region (see Fig. 1). A technique is developed to determine the geometry of energy functionals in such Schrödinger–Kirchhoff-type systems with concave–convex terms and variable exponents.