An $L^1$-theory for $p$-Schr\"odinger equations with confinement in measure

TL;DR

This paper develops an $L^1$-theory for $p$-Schr"odinger equations with confining potentials, introducing asymptotic energy solutions and a new compactness theorem for Sobolev functions in asymptotic $L^p$ spaces.

Contribution

It introduces a novel compactness theorem and establishes existence and uniqueness of solutions for degenerate $p$-Schr"odinger equations with confining potentials.

Findings

Established existence and uniqueness of asymptotic energy solutions for $p eq 2$.

Proved a new Rellich–Kondrachov-type compactness theorem independent of dimension.

Showed solutions coincide with weak solutions in the duality regime.

Abstract

We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions.