We consider the Lp solvability for divergence and non-divergence form Schrödinger equations with discontinuous coefficients. As an application, we give the global Morrey regularity for divergence and non-divergence form Schrödinger operators with VMO coefficients in a bounded domain.

results. Theorem 7.5. Let 1 < p < ∞, 0 ≤ β < n, Ω = Rn , V (x) ∈ Bn and aij (x) ∈ VMO θ (ρ). Then (i) Assume u ∈ Wβ1,p (Ω), f, g ∈ Lp,β (Ω). There exist positive constants λ0 and C, depending only on p, K, n, θ, δ, δ0 , C0 , l0 and β, such that √ λ Du Lp,β (Ω) +λ u Lp,β (Ω) √ ≤C λ g Lp,β (Ω) +C f Lp,β (Ω) , (7.4) provided that λ ≥ λ0 and, Mu + λu = divg + f in Ω. (7.5) (ii) For any λ > λ0 and f, g ∈ Lp,β (Ω), there exists a unique u ∈ Wβ1,p (Ω) of (7.5) satisfying (7.4). Theorem 7.6.

reverse Hölder inequality ⎛ ⎜ ⎝ ⎞1/q ⎟ V q (y)dy ⎠ 1 |B(x, r)| B(x,r) ⎛ ⎜ ≤C⎝ ⎞ ⎟ V (y)dy ⎠ 1 |B(x, r)| (1.4) B(x,r) holds for every x ∈ Rn and 0 < r < ∞, where B(x, r) denotes the ball centered at x with radius r. In particular, if V is a nonnegative polynomial, then V ∈ B∞ . It is worth pointing out that the Bq class is such that, if V ∈ Bq for some q > 1, then there exists ε > 0, which depends only on n and the constant C in (1.4), such that V ∈ Bq+ε . The study of Schrödinger operator

dy ⎠ ≤ CMp (f )(z). Bk Therefore I4 ≤ CMp (f )(z). Thus, (3.2) holds. The proof is ﬁnished. ✷ ∗ Lemma 3.10. Let 1 < p, v < ∞, aij (x) ∈ BMO θ (ρ). If V (x) is Bn/2 , then there exists a constant C such that 1 |Q| |Dij u(x)|dx ≤ C[aij ]θ inf Mpv (Dij u)(y) + C inf Mp (Lu)(y), y∈Q y∈Q Q where Q = B(x0 , ρ(x0 )). Proof. Let a ¯ij = 1 |Q| Q aij (x)dx. We set L0 = −¯ aij Dij + V . Then −1 −1 Dij u = Dij L−1 Luχ(4Q)c , 0 (L0 uχ4Q ) + Dij L0 [(L0 u − Lu)χ(4Q)c ] + Dij L0 then 1 |Q| |Dij

ρ(x0 ) ) |x0 − y|n (1 + ≤C 2k ρ(x0 )<|y−x0 |<2k+1 ρ(x0 ) |Lu(y)|dy G. Pan, L. Tang / Journal of Functional Analysis 270 (2016) 88–133 118 ⎛ ∞ ⎜ 2−kN ⎝ ≤C k=2 ⎞ p1 1 |B(x0 , 2k+1 ρ(x0 ))| ⎟ |Lu(y)|p dy ⎠ B(x0 ,2k+1 ρ(x0 )) ≤ C inf Mp (Lu)(y). y∈Q Case 2. R < 8ρ(x0 ). In this case, let a ¯ij (R) = 1 |B(¯ x0 ,R)| B(¯ x0 ,R) aij (x)dx. We set L0 = −¯ aij (R)Dij +V . Note that x0 , R8 ) ⊆ B(x0 , 8ρ(x0 )), B(x0 , ρ(x0 )) = ∅, then B(¯ and ρ(x0 ) ∼ ρ(¯ x0 ). By Hölder’s inequality and

here. ✷ Similarly to the proof of Theorems 4.1 and 4.2, using Lemmas 3.12 and 4.1–4.4, we have the following results. ∗ and aij (x) ∈ VMO θ (ρ). Then Theorem 4.3. Let 1 < p < ∞, ω ∈ Ap , V (x) ∈ Bn/2 (i) For any u ∈ Wω2,p (Rn ), λu n Lp ω (R ) + √ λux n Lp ω (R ) + uxx n Lp ω (R ) ≤ C (M + λ)u n Lp ω (R ) provided that λ ≥ λ0 , where λ0 , C are depending only on p, K, n, θ, l0 , C0 , δ and Ap (ω). (ii) For any λ > λ0 = λ0 (p, K, d, δ, η, ω) and f ∈ Lpω (Rn ), there exists a unique