Existence of weak positive solutions in anisotropic Musielak–Orlicz–Sobolev spaces for (μ(·), ν(·))-Laplacian-type Kirchhoff models

Abstract The primary objective of this paper is to study the existence of infinitely many positive weak solutions for a class of double-phase <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mover accent="true"> <m:mi>μ</m:mi> <m:mo stretchy="false">→</m:mo> </m:mover> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mover accent="true"> <m:mi>ν</m:mi> <m:mo stretchy="false">→</m:mo> </m:mover> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (\vec{\mu}(\,\cdot\,),\vec{\nu}(\,\cdot\,)) -Kirchhoff-type problems governed by the following elliptic Kirchhoff equation with Dirichlet boundary conditions: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>N</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mi>ℜ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo maxsize="260%" minsize="260%">(</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="script">𝒰</m:mi> </m:msub> <m:mrow> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:msub> <m:mi>μ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo maxsize="210%" minsize="210%">|</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:msub> <m:mi>s</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mfrac> <m:mo maxsize="210%" minsize="210%">|</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mi>μ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:msub> <m:mi>ν</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow>