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<p>In this paper, we study the fractional Kirchhoff-Choquard equation</p><p><Equation ID="Equa"> <EquationSource Format="TEX">\(\matrix{{M\Big([u]_{s,p}^p + {\varepsilon ^{- N}}\int\limits_{{\mathbb{R}^N}} V (x)|u{|^p}{\rm{d}}x\Big)({\varepsilon ^N}(- \Delta)_p^su + V(x)|u{|^{p - 2}}u)} \cr {= {\mkern 1mu} {\varepsilon ^{\mu - N}}\Big(\int\limits_{{\mathbb{R}^N}} {{{Q(y)F(u(y))} \over {|x - y{|^\mu}}}} {\rm{d}}y\Big)Q(x)f(u(x))\quad {\text{in}}\;{{\mathbb{R}^N}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}}\)</EquationSource> </Equation></p><p>where <i>ε</i> is a positive parameter, <i>N</i> = <i>ps,p</i> ≥ 2, <i>s</i> ∈ (0,1), 0 &lt; <i>μ</i> &lt; <i>N</i>. The Kirchhoff function <i>M</i>(<i>t</i>) = <i>a</i> + <i>bt,a</i> &gt; 0,<i>b</i> &gt; 0, nonlinear function <i>f</i> has the exponential growth, potential functions <i>V</i> and <i>Q</i> are continuous functions satisfying some suitable conditions. Using Ljusternik-Schnirelmann category theory and variational methods, we establish the multiplicity and concentration of positive solutions for small values of the parameter.</p>

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Concentration phenomena and competition effects for fractional Kirchhoff-Choquard equations with exponential growth

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Tahir Boudjeriou,
  • Vicenţiu D. Rădulescu,
  • Thin Van Nguyen

摘要

In this paper, we study the fractional Kirchhoff-Choquard equation

\(\matrix{{M\Big([u]_{s,p}^p + {\varepsilon ^{- N}}\int\limits_{{\mathbb{R}^N}} V (x)|u{|^p}{\rm{d}}x\Big)({\varepsilon ^N}(- \Delta)_p^su + V(x)|u{|^{p - 2}}u)} \cr {= {\mkern 1mu} {\varepsilon ^{\mu - N}}\Big(\int\limits_{{\mathbb{R}^N}} {{{Q(y)F(u(y))} \over {|x - y{|^\mu}}}} {\rm{d}}y\Big)Q(x)f(u(x))\quad {\text{in}}\;{{\mathbb{R}^N}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}}\)

where ε is a positive parameter, N = ps,p ≥ 2, s ∈ (0,1), 0 < μ < N. The Kirchhoff function M(t) = a + bt,a > 0,b > 0, nonlinear function f has the exponential growth, potential functions V and Q are continuous functions satisfying some suitable conditions. Using Ljusternik-Schnirelmann category theory and variational methods, we establish the multiplicity and concentration of positive solutions for small values of the parameter.