A unifying zero-mass equation

We introduce the nonlocal zero-mass equation $$-\mathrm{\Delta }u+{\mathrm{\Phi }}_u(\mathrm{\mid }x\mathrm{\mid })\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=f(u)\text{in }{\mathbb{R}}^N,{\mathrm{\Phi }}_u(r)=a{\int }_r^{\mathrm{\infty }}{\rho }^{-b}{h}_u^{q-1}(\rho )d\rho ,$$ where $h_u(\rho )$ is the mass of $\mathrm{\mid }u{\mathrm{\mid }}^p$ in the ball of radius $\rho$. 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 $\lambda >0$ 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 $N\ge 4$.