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September 10, 2025

Normalized solutions to Sobolev critical Schrödinger equations with potential V(x)

Key Points

Normalized solutions exist for the Schrödinger equation under various potential conditions, enhancing previous findings.
The study addresses splittings between the existence of solutions and sobolev critical conditions across different potentials.
Results validate earlier works while extending applicability to a broader class of conditions in the Sobolev critical case.
The focus on normalizations and conditions provides a deeper insight into nonlinearity in the Schrödinger framework.

Abstract

In this paper, we study normalized solutions of the Schrödinger equation with Sobolev critical nonlinearity −Δu−V(x)u=λu+|u|2*−2uu∈H1(RN),λ∈R,∫RNu2=ρ2, where N ≥ 3 and 2*≔2NN−2. We prove existence of normalized solutions of the problem under different conditions on the potential V(x). Our results extend some results of Bartsch et al. Commun. Partial Differ. Equations 46, 1729–1756 (2021) and Molle et al. J. Differ. Equations 333, 302–331 (2022) to the Sobolev critical case.

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Normalized solutions to Sobolev critical Schrödinger equations with potential V(x)

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