A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order. (24th August 2020) - Zuo, Jiabin; An, Tianqing; Fiscella, Alessio
Abstract : The paper deals with the following Kirchhoff‐type problem M ∬ ℝ 2 N 1 p ( x, y ) | v ( x ) − v ( y ) | p ( x, y ) | x − y | N + p ( x, y ) s ( x, y ) d x d y ( − Δ ) p ( · ) s ( · ) v ( x ) = μ g ( x, v ) + | v | r ( x ) − 2 v in Ω, v = 0 in ℝ N Ω, where M models a Kirchhoff coefficient, ( − Δ ) p ( · ) s ( · ) is a variable s (·)‐order p (·)‐fractional Laplace operator, with s ( · ) : ℝ 2 N → ( 0, 1 ) and p ( · ) : ℝ 2 N → ( 1, ∞ ) . Here, Ω ⊂ ℝ N is a bounded smooth domain with N > p ( x, y ) s ( x, y ) for any ( x, y ) ∈ Ω ¯ × Ω ¯, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r ( x ) could be close to the critical exponent p s ∗ ( x ) = N p ¯ ( x ) / ( N − s ¯ ( x ) p ¯ ( x ) ), given with p ¯ ( x ) = p ( x, x ) and s ¯ ( x ) = s ( x, x ) for x ∈ Ω ¯ . 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.