A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

Abstract

The paper deals with the following Kirchhoff‐type problem where M models a Kirchhoff coefficient, is a variable s(·)‐order p(·)‐fractional Laplace operator, with and . Here, is a bounded smooth domain with N > p(x, y)s(x, y) for any , μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent , given with and for . We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.