A critical Kirchhoff‐type problem driven by a <i>p</i> (·)‐fractional Laplace operator with variable <i>s</i> (·) ‐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 &gt; 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.

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