<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential
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<tex-math>\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m},</tex-math>
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where <inline-formula>
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<tex-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}}}</tex-math>
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</inline-formula>. Based on variational approach and a variant of the quantitative strain lemma, for each <italic>b ></italic> 0, we show the existence of a least energy nodal solution <italic>ub</italic>. In addition, a convergence property of <italic>ub</italic> as <italic>b</italic> ↘ 0 is established.
</abstract>

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential



(1+b[ u ]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,
\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m},


where 


[ u ]α=(∫ℝN(| (-Δx)α2u |2+| ∇yu |2)dxdy)12
{\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 ub. In addition, a convergence property of ub as b ↘ 0 is established.

Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation

Oriental Applied Mathematics Laboratory(LAMO)

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential



(1+b[ u...

Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation

Published Online: May 28, 2022

Page range: 212 - 227

Received: Jul 02, 2021

Accepted: Feb 10, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0015

Keywords
Kirchhoff equations, BO-ZK operator, Sign-changing solutions, Deformation lemma

© 2022 Mohammed Rahmani et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation

Published Online: May 28, 2022

Page range: 212 - 227

Received: Jul 02, 2021

Accepted: Feb 10, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0015

KeywordsKirchhoff equations, BO-ZK operator, Sign-changing solutions, Deformation lemma

© 2022 Mohammed Rahmani et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Kirchhoff equations, BO-ZK operator, Sign-changing solutions, Deformation lemma