A Brezis-Nirenberg type result for mixed local and nonlocal operators

TL;DR

The work addresses critical elliptic problems driven by a mixed local–nonlocal operator $\mathcal{L}=-\Delta + (-\Delta)^s$ with $s\in(0,1)$ on open sets, focusing on sharp Sobolev-type inequalities and Brezis–Nirenberg–type existence phenomena. It proves the mixed Sobolev constant $\mathcal{S}_{n,s}(\Omega)$ equals the classical constant $\mathcal{S}_n$ but is never achieved in $\mathcal{X}^{1,2}(\Omega)$, and develops a variational theory for the critical problem $\mathcal{L}u = u^{2^*-1}+\lambda u^p$, $p\in[1,2^*-1)$, including a linear case with an intermediate spectral interval and a superlinear case with a Mountain Pass framework. The analysis reveals how the nonlocal component alters existence regimes compared to the purely local Brezis–Nirenberg problem, including a dichotomy based on the parameters $\kappa_{s,n}=\min\{2-2s,n-2\}$ and $\beta_{p,n}=n-\frac{(p+1)(n-2)}{2}$, and demonstrates how large $\lambda$ or spectral gaps can guarantee solutions. Together, these results extend Brezis–Nirenberg type phenomena to mixed-order operators and provide a foundational variational framework for critical mixed local–nonlocal PDEs.

Abstract

We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. In particular, we investigate the corresponding Sobolev inequality, detecting the optimal constant, which we show that is never achieved. Moreover, we present an existence (and nonexistence) theory for the corresponding subcritical perturbation problem.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Remark 2.2
• Remark 3.1
• proof : Proof of Theorem \ref{['thm:SnsSn']}
• proof : Proof of Theorem \ref{['thm:Snneverachieved']}