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- W3165183235 abstract "Abstract In this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ textstylebegin{cases} (1+ bint _{mathbb{R}^{3}}int _{mathbb{R}^{3}} frac{ vert u(x)-u(y) vert ^{2}}{ vert x-y vert ^{3+2s}},mathrm{d}x ,mathrm{d}y )(-Delta )^{s} u+V(x)u = f(x)u^{-gamma }, & x in mathbb{R}^{3}, u>0,& xin mathbb{R}^{3}, end{cases} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:msub> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:msub> <mml:mfrac> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $(-Delta )^{s}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:math> is the fractional Laplacian with $0< s<1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>s</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:math> , $b>0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>b</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> is a constant, and $gamma >1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:math> . Since $gamma >1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>γ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:math> , the energy functional is not well defined on the work space, which is quite different with the situation of $0<gamma <1$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>γ</mml:mi> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> </mml:math> and can lead to some new difficulties. Under certain assumptions on V and f , we show the existence and uniqueness of a positive solution $u_{b}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> by using variational methods and the Nehari manifold method. We also give a convergence property of $u_{b}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> as $brightarrow 0$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>b</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:math> , where b is regarded as a positive parameter." @default.
- W3165183235 created "2021-06-07" @default.
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- W3165183235 date "2021-03-19" @default.
- W3165183235 modified "2023-10-14" @default.
- W3165183235 title "Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity" @default.
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- W3165183235 doi "https://doi.org/10.1186/s13661-021-01507-8" @default.
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