Lane-Emden equation for the superposition of a local and nonlocal operator with the "wrong sign"

We consider the Lane-Emden equation associated with the operator $-Δ- γ\, (-Δ)^s$ (for positive $γ$). First we consider the equation in the entire space and we prove that no Serrin-type critical exponent arises in the $H^1(\mathbb R^n)$ framework. Secondly we consider the problem settled in an open, bounded and connected set and we prove the existence of a positive solution for sufficiently small $s$.