A unifying zero-mass equation
Abstract
We introduce the nonlocal zero-mass equation − Δu + Φu(|x|) |u|p−2 u = f(u) in ℝN, Φu(r) = a ∫r∞ ρ−b huq−1(ρ) dρ, where hu(ρ) is the mass of |u|p in the ball of radius ρ. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every λ> 0 that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions N ≥ 4.
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BibTeX
@misc{perera2026unifying,
title = {{A unifying zero-mass equation}},
author = {Kanishka Perera},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26585},
eprint = {2609.26585},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26585},
}
RIS
TY - UNPB
TI - A unifying zero-mass equation
AU - Kanishka Perera
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26585
UR - https://arxiv.org/abs/2609.26585
AB - We introduce the nonlocal zero-mass equation − Δu + Φ_u(|x|) |u|^p−2 u = f(u) in ℝ^N, Φ_u(r) = a ∫_r^∞ ρ^−b h_u^q−1(ρ) dρ, where h_u(ρ) is the mass of |u|^p in the ball of radius ρ. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every λ> 0 that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions N ≥ 4.
ER -