Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential
Junwei Yu
TL;DR
This work investigates the existence and multiplicity of normalized standing waves for the energy-critical Schrödinger equation with a trapping potential in 𝑅^N (N≥3). It adopts a constrained variational framework on the mass sphere by transforming to u with μ = ρ^{2^*-2} and studying the energy E_μ on 𝓜, proving the existence of a positive local minimizer (ground state) and, under the same hypotheses, a mountain-pass solution for small mass. The analysis overcomes compactness challenges from the Sobolev critical exponent using Pohozaev identities, scaling arguments, and Struwe-type monotonicity/compactness techniques within a trapping potential that grows at infinity. The results extend normalized-solution theory to Sobolev-critical regimes with trapping potentials, providing precise energy-level estimates and establishing positivity of the solutions via the maximum principle.
Abstract
We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schrödinger equation $-ΔU + V(x) U = λU + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N} U^2\,dx = ρ^2$, where $N \ge 3$, $V\ge 0$ is a trapping potential, $λ\in \mathbb{R}$ and $2^*=\frac{2N}{N-2}$. Our first result is that the existence of local minimum solutions for $ρ\in (0, ρ^*)$, for some suitable $ρ^* > 0$, under appropriate assumptions on the potential. These solutions correspond to ground states. Our second result concerns the existence of mountain pass solutions, under the same assumptions.