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

The paper deals with the following Kirchhoff-type problem (Formula presented.) where M models a Kirchhoff coefficient, (Formula presented.) is a variable s(·)-order p(·)-fractional Laplace operator, with (Formula presented.) and (Formula presented.). Here, (Formula presented.) is a bounded smooth domain with N > p(x, y)s(x, y) for any (Formula presented.), μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent (Formula presented.), given with (Formula presented.) and (Formula presented.) for (Formula presented.). 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.