On a elliptic problem involving variable-order fractional $p(\cdot)-$ Laplacian and logarithmic nonlinearity

Document Type : Letters to the editor

Authors

1 Laboratoire de Mathématiques, Informatique et Applications, Université Norbert ZONGO, Koudougou, Burkina Faso

2 Laboratoire de Mathématiques, Informatique et Applications, Université Norbert ZONGO, BP 376 Koudougou, Burkina Faso

3 Département de Mathématiques et Informatique, Université Lédéa Bernard OUEDRAOGO, BP 346, Ouahigouya, Burkina Faso

10.21608/jfca.2025.383385.1175

Abstract

This paper investigates the existence of weak solutions for a fractional elliptic problem with variable exponent and variable order, using Ekeland's variational principle. The equation studied involves the generalised fractional Laplacian operator, denoted $(- \Delta)_{p(\cdot)}^{s(\cdot)}$, specialised in modelling complex real or physical phenomena, where p and s are continuous functions of real variables with values in $(0, \infty)$ and $(0, 1)$, respectively. The method is based on the variational formulation associated with the fractional elliptic equation. We consider a functional for which a minimizer is sought in a fractional Sobolev space. Under certain assumptions on the exponents and the order of derivation, we have shown that this functional admits a minimizer. This minimizer is a weak solution of the elliptic equation.
This approach makes it possible to treat non-local problems with variable exponents and order of derivation, thus offering an extension of the classical results to more complex cases. The functional setting involves Lebesgue and Sobolev spaces with variable exponent and variable-order.

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