On the behavior of least energy solutions of a fractional ( p , q ( p ) )-Laplacian problem as p goes to infinity

Abstract

We study the behavior as p → ∞ of u p , a positive least energy solution of the problem [ ( − Δ p ) α + ( − Δ q ( p ) ) β ] u = μ p | u ( x u ) | p − 2 u ( x u ) δ x u in  Ω u = 0 in  R N ∖ Ω | u ( x u ) | = ‖ u ‖ ∞ , where Ω ⊂ R N is a bounded, smooth domain, δ x u is the Dirac delta distribution supported at x u , lim p → ∞ q ( p ) p = Q ∈ ( 0 , 1 ) if  0 < β < α < 1 ( 1 , ∞ ) if  0 < α < β < 1 and lim p → ∞ μ p p > R − α , with R denoting the inradius of Ω.