Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities

Colin Fried; Mathew Gluck

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arXiv · 2607.27423

Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities

Colin Fried·Mathew Gluck

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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.

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Colin Fried, Mathew Gluck. 2026-07-29. Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities. https://arxiv.org/abs/2607.27423

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