Trudinger–Moser‐type inequality with logarithmic convolution potentials

Abstract

We establish Moser-Trudinger-type inequalities in the presence of a logarithmic convolution potential when the domain is a ball or the entire space ℝ 2 . Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler-Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations.

A log-weighted Moser inequality on the plane

“…Note that, thanks to ( 12) and ( 13), the scaling factor appearing when changing variable in the Dirichlet norm has an interesting property: (17) rT (r)…”

“…For this reason, these kind of inequalities have been established, till now, only in the framework of radial Sobolev spaces (see, e.g., [23,25,29] and references therein). Up to our knowledge, even in the subcritical Trudinger setting, only Albuquerque [4] and the author [36] have addressed the question in the general framework of mass weighted Sobolev spaces, that is, without any a-priori restriction to radial functions (see also [17] for a close result).…”

Nonlocal Planar Schrödinger-Poisson Systems in the Fractional Sobolev Limiting Case

“…In this setting, if f (u) = u, an approach originating from the unpublished work of Stubbe [St] was proposed in [CW,DW,BCV,CW2], according to which (Ch) is solved in a constraint subspace of H 1 (R 2 ), where the logarithmic convolution term is well-defined.…”