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 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.