Fractional critical systems with mixed boundary conditions

TL;DR

We address the existence of nontrivial weak solutions for a fractional elliptic system with critical coupling under mixed Dirichlet-Neumann boundary conditions. The authors develop a variational framework for the spectral fractional Laplacian and employ the Caffarelli-Silvestre extension to obtain a local formulation, complemented by an orthogonalization approach to control Palais-Smale sequences under one-sided resonance $\Lambda_k \leq \mu_1 \leq \mu_2 < \Lambda_{k+1}$. A Sobolev-constant analysis, augmented by Aubin-Talenti bubbles and careful test-function estimates, yields a linking geometry in the associated energy space and a Palais-Smale condition below a sharp critical level $c_0$. Under the assumptions $N>8s$ and $\mu_1,\mu_2$ lying in the spectral gap, the paper proves the existence of at least one nontrivial weak solution, extending classical results to fractional mixed-boundary problems and contributing tools for nonlocal vector-valued systems with critical growth.

Abstract

In this paper, we analyze the existence of solution for a fractional elliptic system coupled by critical nonlinearities and endowed with mixed Dirichlet-Neumann boundary conditions. By means of variational methods and an orthogonalization-like process in the corresponding Sobolev space, we establish the existence of at least one weak solution.

Theorems & Definitions (22)

• Remark 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3