On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''
Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi
TL;DR
This work analyzes a Sobolev-critical elliptic problem with a leading operator $- abla^2 - \gamma(-\Delta)^s$ in a bounded domain, treating the fractional term as a nonlocal perturbation with the "wrong sign". A variational framework in the mixed space $\mathcal{X}^{1,2}(\Omega)$ is developed, focusing on the constrained quotient $S(\gamma)=\inf\{\|\nabla u\|^2 - \gamma [u]_s^2 : \|u\|_{L^{2^*}}=1\}$ and its comparison to the best Sobolev constant $S_n$. The authors prove existence of nontrivial weak solutions: for $n\ge5$ this holds for all $\gamma\in(0,C_{emb})$, while for $n=3,4$ there exists a threshold $\gamma^*\in(0,C_{emb})$ such that solutions exist for $\gamma>\gamma^*$. The approach combines sharp energy estimates using Aubin–Talenti bubbles, a Brezis–Lieb decomposition, and continuity/monotonicity properties of $S(\gamma)$ to overcome the lack of scaling invariance and handle the local-nonlocal perturbation.
Abstract
We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. The main novelty is that we consider a mixed operator of the form $-Δ- γ(-Δ)^s$, namely we suppose that the fractional Laplacian has the ``wrong sign'' and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.