An $L^1$-theory for $p$-Schrödinger equations with confinement in measure

We consider stationary $p$-Schrödinger 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 $p\ge 2$. The proof relies on a new Rellich$\text{unicode}x2013$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 $L^1({\mathbb{R}}^n)\cap L^{p^{\mathrm{\prime }}}({\mathbb{R}}^n)$, 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.