Bifurcation from Infinity of the Schrödinger Equation via Invariant Manifolds

The uploaded paper (Li–Wang, 2021) proves, via an invariant-manifold reduction and Conley index/shape theory, that for the stationary Schrödinger equation −Δu+V(x)u=λu+f(x,u) on RN, under (A1)–(A3) and a smallness condition (F) quantified by FμLf<1, there exists θ>0 such that for every λ∈[λ*−θ,λ*) there are at least three distinct nontrivial solutions; two blow up as λ↗λ* while a third remains bounded in a two-sided neighborhood of λ* (Theorem 4.4) . The paper builds a global invariant manifold Mλ as a Lipschitz graph over the finite-dimensional eigenspace Y2 associated with the isolated eigenvalue λ* (Theorem 3.1; condition (F) and its explicit constant Fμ are stated in (4.4)) , reduces the flow to Y2 (equation (4.8)), proves an outward-pointing estimate on large spheres in Y2 derived from a Landesman–Lazer lemma (Lemma 4.2, yielding d/dt|w|2>c0|w| for |w|≫1) , and obtains both an “unbounded” sphere-shaped invariant set S∞λ and a bounded invariant set Kλ across λ* (Theorem 4.3) . The model’s solution follows the same structure: invariant-manifold reduction, finite-dimensional equation (λ*−λ)z = P f(z+hλ(z)), LL-based outward estimate on large spheres, Conley index jump across λ*, and shape theory producing at least two equilibria on the sphere and one bounded equilibrium. A minor difference is that the paper states blow-up qualitatively (min‖w‖ on S∞λ →∞ and Rλ∼C1/(λ*−λ) as λ↗λ*; see (4.16)–(4.17)) , while the model spells out the explicit lower bound |z|≳1/(λ*−λ) for equilibria; this follows directly from the equilibrium identity (λ*−λ)|w|2=(f(w+ξλ(w)),w) (obtained from (4.10) with w′=0) together with the LL-based positivity (a consequence of Lemma 4.2), even though the paper does not highlight the rate explicitly . The model also mentions “unique continuation” (not needed in the paper’s finite-dimensional argument) and slightly overstates that “every equilibrium lies on the manifold” (true for equilibria via the integral formulation but not emphasized in the paper). Overall, the arguments align and are essentially the same proof strategy.