On a class of fractional \(p(x,y)-\)Kirchhoff type problems with indefinite weight

Seyed Mostafa Sajjadi; Ghasem Alizadeh Afrouzi

doi:10.56754/0719-0646.2601.107

DOI:

https://doi.org/10.56754/0719-0646.2601.107

Abstract

This paper is concerned with a class of fractional \(p(x,y)-\)Kirchhoff type problems with Dirichlet boundary data along with indefinite weight of the following form
\begin{equation*}
\left\lbrace\begin{array}{ll}
M\left(\int_{Q}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dx\,dy\right)\\
(-\triangle_{p(x)})^s+|u(x)|^{q(x)-2}u(x) & \\
=\lambda V(x)|u(x)|^{r(x)-2}u(x)& \text{in }\Omega,\\
u=0, & \text{in }\mathbb{R}^N\Omega.
\end{array}\right.
\end{equation*}

By means of direct variational approach and Ekeland’s variational principle, we investigate the existence of nontrivial weak solutions for the above problem in case of the competition between the growth rates of functions \(p\) and \(r\) involved in above problem, this fact is essential in describing the set of eigenvalues of this problem.

Keywords

Kirchhoff type problems , indefinite weight , Ekeland’s variational principle , variable exponent , fractional p(x, y)−Laplacian problems

Mathematics Subject Classification:

35R11 , 35D30 , 35J20 , 46E35

Seyed Mostafa Sajjadi Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. https://orcid.org/0009-0002-2675-1497
Ghasem Alizadeh Afrouzi Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. https://orcid.org/0000-0001-8794-3594

Pages: 107–122
Date Published: 2024-04-08
Vol. 26 No. 1 (2024)

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