Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities

2026/07/29 by Colin Fried, Mathew Gluck
Mathematics · #math.AP #msc:35J50 #msc:35J57 #msc:35J91

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arxiv created 2026/07/29 · arxiv updated 2026/07/31

Abstract

This paper concerns a variational system of nonlinear elliptic equations that generalizes the classical Brezis-Nirenberg problem. In addition to considering vector-valued unknown functions in place of scalar-valued unknown functions, the problem under consideration generalizes the Brezis-Nirenberg problem in two directions. First, the nonlinearity is the sum of two homogeneous functions of suitable homogeneity degrees. Second, the Sobolev-critical term in the nonlinearity is nonlocal. We establish conditions under which a nontrivial solution to the system under consideration is guaranteed to exist and conditions under which the system under consideration admits only the trivial solution.

Citations

• Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

Related

• The Brezis-Nirenberg problem for nonlocal systems
• The quasi-linear Brezis-Nirenberg problem in low dimensions
• Nonlocal problems with Hardy-Littlewood-Sobolev critical exponent and Hardy potential
• Nontrivial solutions for semilinear elliptic systems via Orlicz-Sobolev theory
• Critical equations with a sharp change of sign in the nonlinearity