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• W4223991173 abstract "Abstract In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msubsup> <m:mrow> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mi>u</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=normal>Δ</m:mi> </m:mrow> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>α</m:mi> </m:msup> <m:mi>u</m:mi> <m:mo>-</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=normal>Δ</m:mi> </m:mrow> <m:mi>y</m:mi> </m:msub> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mi>N</m:mi> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> <m:mo>,</m:mo> </m:mrow> </m:math> left( {1 + bleft[ u right]_alpha ^2} right)left( {{{left( { - {Delta _x}} right)}^alpha }u - {Delta _y}u} right) + Vleft( {x,y} right)u = fleft( {x,y,u} right),left( {x,y} right) in {mathbb{R}^N} = {mathbb{R}^n} times {mathbb{R}^m}, where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=inline> <m:mrow> <m:msub> <m:mrow> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mi>u</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>α</m:mi> </m:msub> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>∫</m:mo> <m:mrow> <m:msub> <m:mrow /> <m:mrow> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mrow> <m:mi mathvariant=normal>Δ</m:mi> </m:mrow> <m:mi>x</m:mi> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mfrac> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>y</m:mi> </m:msub> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>d</m:mi> <m:mi>x</m:mi> <m:mi>d</m:mi> <m:mi>y</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:math> {left[ u right]_alpha } = {left( {int {_{{mathbb{R}^N}}left( {{{left| {{{left( { - {Delta _x}} right)}^{{alpha over 2}}}u} right|}^2} + {{left| {{nabla _y}u} right|}^2}} right)dxdy} } right)^{{1 over 2}}} . Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution u b . In addition, a convergence property of u b as b ↘ 0 is established." @default.
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• W4223991173 date "2022-05-01" @default.
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• W4223991173 title "Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation" @default.
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• W4223991173 doi "https://doi.org/10.2478/mjpaa-2022-0015" @default.
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