Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 83, pp. 1–26. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.83 HÖLDER REGULARITY OF WEAK SOLUTIONS TO NONLOCAL p-LAPLACIAN TYPE SCHRÖDINGER EQUATIONS WITH Ap 1-MUCKENHOUPT POTENTIALS YONG-CHEOL KIM Abstract. In this article, using the De Giorgi-Nash-Moser method, we obtain an interior Hölder continuity of weak solutions to nonlocal p-Laplacian type Schrödinger equations given by an integro-differential operator Lp K (p > 1), Lp Ku+ V |u|p−2u = 0 in Ω, u = g in Rn\Ω. Where V = V+ − V− with (V−, V+) ∈ L1 loc(R n) × Lq loc(R n) for q > n ps > 1 and 0 < s < 1 is a potential such that (V−, V b,i + ) belongs to the (A1, A1)-Muckenhoupt class and V b,i + is in the A1-Muckenhoupt class for all i ∈ N . Here, V b,i + := V+ max{b, 1/i}/b for an almost everywhere positive bounded function b on Rn with V+/b ∈ Lq loc(R n), g ∈ W s,p(Rn) and Ω ⊂ Rn is a bounded domain with Lipschitz boundary. In addition, we prove local boundedness of weak subsolutions of the nonlocal p-Laplacian type Schrödinger equations. Also we obtain the logarithmic estimate of the weak supersolutions which play a crucial role in proving Hölder regularity of the weak solutions. We note that all the above results also work for a nonnegative potential in Lq loc(R n) (q > n ps > 1, 0 < s < 1). 1. Introduction The research on nonlocal partial differential equations has been performed not only in pure mathematics, but also in scientific areas that necessitate its concrete applications. This kind of problems appear in various applications such as continuum mechanics, phase transition phenomena related to a nonlocal version of classical Allen-Cahn equation, population dynamics, nonlocal minimal surfaces, a nonlocal version of Schrödinger equations for standing waves (see [3, 5, 6]), game theory and also constrained variational problems with fractional diffusion arising in the quasi-geostrophic flow model, anomalous diffusions and American options with jump processes (see [7, 8, 31]). For q > n ps > 1 (p > 1, 0 < s < 1), let Ps,p q (Rn) be the class of potentials V = V+ − V− such that (i) V− ∈ L1 loc(Rn), (ii) V+ ∈ Lq loc(Rn), (iii) there is an almost everywhere positive bounded function b on Rn so that V+/b ∈ Lq loc(Rn), (V−, V b,i + ) belongs to the (A1, A1)-Muckenhoupt class and V b,i + is in the A1-Muckenhoupt class for all i ∈ N, where V b,i + := max{b, 1/i} b V+. 2020 Mathematics Subject Classification. 47G20, 45K05, 35J60, 35B65, 35D10, 60J75. Key words and phrases. Hölder regularity; nonlocal p-Laplacian type Schrödinger equation; Ap 1-Muckenhoupt potential; De Giorgi-Nash-Moser method. ©2025. This work is licensed under a CC BY 4.0 license. Submitted February 21, 2025. Published August 8, 2025. 1 2 Y.-C. KIM EJDE-2025/83 If V ∈ Ps,p q (Rn) for q > n ps > 1 ( p > 1, 0 < s < 1), then we say that V is a Ap 1-Muckenhoupt potential. When p = 2, we call it A1-Muckenhoupt potential. The aim of this paper is to obtain an interior Hölder regularity of weak solutions of nonlocal p- Laplacian type Schrödinger equations with Ap 1-Muckenhoupt potentials and to additionally obtain the local boundedness of weak subsolutions of the nonlocal equation. For p > 1, let Kp be the family of all positive symmetric kernels satisfying the uniformly ellipticity assumption cn,p,sλ |y|n+ps ≤ K(y) = K(−y) ≤ cn,p,sΛ |y|n+ps , 0 < s < 1, y ∈ Rn\{0}, (1.1) where cn,p,s > 0 is the normalization constant given by cn,p,s = Γ(n+p 2 ) p (1− s) π n−1 2 Γ(p+1 2 ) . For K ∈ Kp (p > 1), we consider integro-differential operators Lp K given by Lp Ku(x, t) = p. v. ∫ Rn Hp(u(x)− u(y))K(x− y) dy (1.2) where Hp(t) = |t|p−2t for t ∈ R. If p = 2, then we write Lp K = LK . In particular, if K(y) = cn,p,s|y|−n−ps, then Lp K = (−∆)sp is the fractional p-Laplacian and it is well-known [20] that lim s→1− (−∆)spu = −∆pu for any function u in the Schwartz space S(Rn), where ∆pu = div(|∇u|p−2∇u) is the classical p-Laplacian. We are interested in the Dirichlet problem for the nonlocal p-Laplacian type Schrödinger equa- tion Lp Ku+ V |u|p−2u = 0 in Ω u = g in Rn\Ω (1.3) where V ∈ Ps,p q (Rn) for q > n ps > 1 (p > 1 and 0 < s < 1), g ∈ W s,p(Rn) and Ω ⊂ Rn is a bounded domain with Lipschitz boundary. The existence and uniqueness of weak solution to the above nonlocal equation was obtained in [24] by applying standard technique of calculus of variations. More precisely speaking about the problem, by employing the De Giorgi-Nash-Moser theory we obtain an interior Hölder continuity of weak solutions to the nonlocal p-Laplacian type Schrödinger equations with Ap 1-Muckenhoupt potentials, and we obtain the local boundedness of weak subsolutions of the nonlocal equation. Here, we note that the boundary condition is imposed on Rn\Ω with nonlocality. In fact, from the probabilistic point of view, it conforms to the natural phenomenon that a discontinuous Lévy process on the domain Ω can exit Ω for the first time jumping to any point in Rn\Ω. When p = 2, the research on the nonlocal equations was strongly motivated by the study of standing wave solutions of the form Ψ(x, t) = e−iωtu(x) of the time-dependent nonlocal Schrödinger equations i ∂Ψ ∂t = LKΨ+ V (x)Ψ which is a fundamental equation of fractional quantum mechanics and fractional quantum physics. This equation was used for the first time in the literature by Laskin [25]. It turns out in Section 3 that any potential V in Ps,p q (Rn) satisfies∫ Rn |φ(y)|V−(y) dy ≤ ∫ Rn |φ(y)|V+(y) dy , (1.4) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 3∫ Rn |φ(y)|p V−(y) dy ≤ ∫ Rn |φ(y)|p V+(y) dy (1.5) for every φ ∈ Y s,p 0 (Ω), whenever q > n ps > 1 (p > 1 and 0 < s < 1). As a matter of fact, the inequality (1.4) is a useful tool for the proof of nonlocal Caccioppoli type inequality to be given in Theorem 1.5, and also the inequality (1.5) makes it possible to prove in Lemma 2.3 below that Y s,p 0 (Ω) is a quasi-Banach space. Notation. • For r > 0, x0 ∈ Rn and s ∈ (0, 1), let us denote by B0 r = Br(x0), Br = Br(0). • For two quantities a and b, we write a ≲ b (resp. a ≳ b) if there is a universal constant C > 0 (depending only on λ,Λ, n, p, s and Ω) such that a ≤ Cb (resp. b ≤ Ca). • For a, b ∈ R, we denote by a ∨ b = max{a, b} and a ∧ b = min{a, b}. • Let Fn be the family of all real-valued Lebesgue measurable functions on Rn. • For u ∈ C(B0 r ), we consider the norm ∥u∥C(B0 r) = sup x∈B0 r |u(x)|. For γ ∈ (0, 1), the γth Hölder seminorm of u on B0 r is defined by [u]Cγ(B0 r) = sup x,y∈B0 r , x ̸=y |u(x)− u(y)| |x− y|γ and the γth Hölder norm of u on B0 r is defined by ∥u∥Cγ(B0 r) = ∥u∥C(B0 r) + [u]Cγ(B0 r) . • For x0 ∈ Ω, p > 1 and r > 0 with B0 r ⊂ Ω, the nonlocal tails of the function u in B0 r ⊂ Ω is defined by Tr(u;x0) = ( rps ∫ Rn\Br(x0) |u(y)|p−1 |y − x0|n+ps dy ) 1 p−1 . (1.6) We now state one of our main results which is called the local boundedness of weak subsolutions to the nonlocal p-Laplacian type Schrödinger equation (1.3), as follows. Theorem 1.1. Let V ∈ Ps,p q (Rn), g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, s ∈ (0, 1)) and B0 2r ⊂ Ω. If u ∈ Y s,p g (Ω)− is a weak subsolution of nonlocal p-Laplacian type Schrödinger equation (1.3), then there is a constant C0 > 0 depending only on n, s, p, λ,Λ and Ω such that sup B0 r u ≤ δ Tr(u+;x0) + C0 δ − (p−1)n sp2 ( − ∫ B0 2r up + dx )1/p for all δ ∈ (0, 1]. 1.1. Remarks. (a) If u ∈ Y s,p g (Ω)− is a weak subsolution of the nonlocal p-Laplacian type Schrödinger equation (1.3) and g ∈ W s,p(Rn) for s ∈ (0, 1), then we see that u ∈ Lp(Ω) and u ≤ g on Rn\Ω, and thus u+ ≤ g+ there. Then it follows from Hölder’s inequality and fractional Sobolev inequalities (2.4), (2.5) below that [Tr(u+;x0)] p−1 ≤ rps (∫ Rn\Ω gp−1 + (y) |y − x0|n+ps dy + ∫ Ω\Br(x0) |u(y)|p−1 |y − x0|n+ps dy ) ≲ r−n(1− 1 p ) ((p− 1)n+ p2s)1/p ( ∥g∥p−1 W s,p(Rn) + ∥u∥p−1 Lp(Ω) ) < ∞. (b) If u ∈ Y s,p g (Ω)+ is a weak supersolution of the nonlocal p-Laplacian type Schrödinger equation (1.3) and g ∈ W s,p(Rn) for s ∈ (0, 1), then −u is its weak subsolution and u ≥ g on Rn\Ω, and so u− ≤ g− there. Then, as in the above (a), we obtain that [Tr(u−;x0)] p−1 ≲ r−n(1− 1 p ) ((p− 1)n+ p2s)1/p ( ∥g∥p−1 W s,p(Rn) + ∥u∥p−1 Lp(Ω) ) < ∞. 4 Y.-C. KIM EJDE-2025/83 Then, from Theorem 1.1, we easily have − inf B0 r u = sup B0 r (−u) ≤ δTr(u−;x0) + C0 δ − (p−1)n sp2 ( − ∫ B0 2r up − dx )1/p for all δ ∈ (0, 1]. (c) If u ∈ Y s,p g (Ω)+ is a weak solution of the nonlocal p-Laplacian type Schrödinger equation (1.3) and g ∈ W s,p(Rn) for s ∈ (0, 1), then it follows from (a), (b) and Theorem 1.1 that oscB0 r u ≤ 2δ Tr(u;x0) + 2C0 δ − (p−1)n sp2 ( − ∫ B0 2r |u|p dx )1/p for all δ ∈ (0, 1]. The next logarithmic estimate plays a crucial role in proving Hölder regularity of weak solutions to the nonlocal p-Laplacian type Schrödinger equation and in showing that the logarithm of its weak solution is a function with locally bounded mean oscillation. In a different way from that of [10], we obtain the logarithmic estimate. Theorem 1.2. Let V ∈ Ps,p q (Rn) and g ∈ W s,p(Rn) for q > n ps > 1 (p > 1 and 0 < s < 1). If u ∈ Y s,p g (Ω)+ is a weak supersolution of nonlocal p-Laplacian type Schrödinger equation (1.3) with u ≥ 0 in B0 R ⊂ Ω, then there is a constant c0 > 0 depending only on n, s, p, λ,Λ and Ω such that∫∫ B0 r×B0 r ∣∣∣ ln(u(x) + b u(y) + b )∣∣∣p dK(x, y) ≤ c0r n−ps [ 1 bp−1 ( r R )ps [TR(u−;x0)] p−1 + ( 1 + ∥V+∥Lq(Ω) )] for any b ∈ (0, 1) and r ∈ (0, R/2), where dK(x, y) = K(x− y) dx dy. Employing the De Giorgi-Nash-Moser theory and using Theorems 1.1 and 1.2, we obtain the fol- lowing Hölder continuity of weak solutions to the nonlocal p-Laplacian type Schrödinger equation, and also we can easily derive Corollary 1.4 as a natural by-product of Theorem 1.3. Theorem 1.3. Let V ∈ Ps,p q (Rn), g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, 0 < s < 1), and let B0 2R ⊂ Ω. If u ∈ Y s,p g (Ω) is a weak solution of the nonlocal p-Laplacian type Schrödinger equation (1.3), then there exist constants η−0 ∈ (0, ps 2(p−1) ) and η+0 ∈ ( ps 2(p−1) , ps p−1 ) such that u is locally η-Hölder continuous in Ω for any η ∈ (0, η−0 ] ∪ [η+0 , ps p−1 ). Furthermore, for each x0 ∈ Ω and for each η ∈ (0, η−0 ] ∪ [ η+0 , ps p−1 ) , we have oscB0 r u ≲ ( r R )η[ TR(u;x0) + ( − ∫ B0 2R |u(x)|p dx )1/p] (1.7) for any r ∈ (0, R/2). Here it turns out that there exist universal constants c0, c∗ > 0 such that η±0 = ln ( 1± √ 1−4 δ ps p−1 2 ) ln δ for δ = e−(c0/c∗)(1+∥V+∥Lq(Ω)) 1/p ∧ (1 4 ) p−1 ps . The next corollary can easily be obtained by using Theorem 1.3 and employing the interpolation on Hölder spaces between Cη− 0 (B0 r ) and Cη+ 0 (B0 r ), which eventually fill up an interior η-Hölder continuity of u in Ω for all η ∈ (η−0 , η + 0 ). Corollary 1.4. Let V ∈ Ps,p q (Rn), g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, 0 < s < 1), and let B0 2R ⊂ Ω. If u ∈ Y s,p g (Ω) is a weak solution of the nonlocal p-Laplacian type Schrödinger equation (1.3), then we have the estimate sup r∈(0,R/2) ∥u∥Cη(B0 r) ≲ 1 Rη [ TR(u;x0) + ( − ∫ B0 2R |u(x)|p dx )1/p] (1.8) for all η ∈ ( 0, ps p−1 ) . If p−1 p < s < 1, then we can expect the better regularity, i.e. C1,α-estimate for some α ∈ (0, 1). As a basic tool for our main results, we show that any weak subsolution of the nonlocal p-Laplacian type Schrödinger equations enjoys the following nonlocal Caccioppoli type inequality. EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 5 Theorem 1.5. Let V ∈ Ps,p q (Rn), g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, s ∈ (0, 1)), and let B0 2r ⊂ Ω. If u ∈ Y s,p g (Ω)− is a weak subsolution of the nonlocal p-Lapacian type Schrödinger equation (1.3), then for any nonnegative ζ ∈ C∞ c (B0 r ) we have the estimate∫ B0 r [w(y)ζ(y)]pV (y) dy + ∫∫ B0 r×B0 r |ζ(x)w(x)− ζ(y)w(y)|p dK(x, y) ≤ 22p+1 (1 4 + cp ) ∫∫ B0 r×B0 r [w(x) ∨ w(y)]p|ζ(x)− ζ(y)|p dK(x, y) + 2p+2 ( sup x∈supp(ζ) ∫ Rn\B0 r wp−1(y)K(x− y) dy ) ∥wζp∥L1(B0 r) where w = (u−M)+ for M ∈ (0,∞) and cp = 1 2 [2(p− 1)]p−1. Remark 1.6. (a) In case that p = 2 and s = 1, the study of the classical Schrödinger operator, i.e. local Schrödinger operator −∆+ V has been ongoing actively and widely in analysis area in Mathematics and Mathematical Physics (refer to [1, 4, 12, 29, 30, 32]). (b) When p = 2 and V ∈ Lq loc(Rn) with q > n 2s (0 < s < 1) is nonnegative, it is known in [5] that a fundamental solution for nonlocal Schrödinger operator LK +V exists and its decay can be obtained. Under an additional restiction that the potential V is in a reverse Hölder class RHγ for γ > n 2s > 1 (0 < s < 1), the Lα −Lβ estimate for the Schrödinger operator LK + V was obtained inside certain trapezoidal region Z which is consist of ( 1 α , 1 β ) and also the weak type Lα − Lβ estimate was partially obtained on the boundary of the region Z (see [6]). (c) In case that p = 2, 0 < s < 1 and V is an A1-Muckenhoupt potential, it was shown in [24] that a fundamental solution for nonlocal Schrödinger operator LK + V exists and its decay can be obtained. Moreover, Hölder continuity and nonlocal Harnack inequalities for LK + V were obtained in [22] and [23]. (d) When V = 0 and 0 < s < 1, the nice result of this problem was obtained by Di Castro, Kuusi and Palatucci [9]; as a matter of fact, when p ∈ (1,∞), they proved nonlocal Harnack inequalities for elliptic nonlocal p-Laplacian equations there. Also they obtained Hölder regularity in [10]. (e) In case that p = 2 and 0 < s < 1, nonlocal Harnack inequalities for (locally nonnegative in Ω) weak solutions of nonlocal heat equations was obtained in [21] by applying the De Giorgi- Nash-Moser theory and the Krylov-Safonov covering theorem [27]. (f) When the nonlocal equation (1.3) with the forcing term f ∈ L∞(Ω) and V = g = 0 is considered for 0 < s < 1 < p < ∞ and a bounded domain Ω ⊂ Rn with C1,1 boundary, Iannizzotto, Mosconi and Squassina obtained the first global Hölder regularity for its weak solutions in [19], i.e. there exist some α ∈ (0, s] and C > 0 depending only on n, p, s and Ω such that ∥u∥Cα(Ω) ≤ C ∥f∥ 1 p−1 L∞(Ω) for any weak solution u ∈ W s,p 0 (Ω) of the nonlocal equation. If the nonlocal equation mentioned just in the above is considered for 0 < s < 1, p ≥ 2 and a bounded domain Ω ⊂ Rn with C1,1 boundary, then they also established the first fine boundary regularity for its weak solutions in [18], i.e. there exist some α ∈ (0, s] and C > 0 depending only on n, p, s and Ω such that∥∥∥ u dsΩ ∥∥∥ Cα(Ω) ≤ C∥f∥ 1 p−1 L∞(Ω) for any weak solution u ∈ W s,p 0 (Ω) of the nonlocal equation, where dΩ(x) = dist(x, ∂Ω). The article is organized as follows. In Section 2, we furnish the function spaces and the definition of weak solutions of the nonlocal p-Laplacian type Schrödinger equation given in (1.3), and mention a well-known lemma which is very useful in applying the De Giorgi-Nash-Moser theory. In Section 3, we give a brief introduction about weighted norm inequalities and the Ap-Muckenhoupt class. Additionally, we furnish several examples about sign-changing potentials which is in the class Ps,p q (Rn) (q > n ps > 1, p > 1, 0 < s < 1). In Section 4, we obtain a sort of nonlocal Caccioppoli type inequality and several useful local properties of weak solutions to the nonlocal p-Laplacian 6 Y.-C. KIM EJDE-2025/83 type Schrödinger equation by using it. In Section 5, we show that the logarithm of a weak solution to the nonlocal p-Laplacian type Schrödinger equation with Ap 1-Muckenhoupt potentials becomes a function with locally bounded mean oscillation. In Section 6, we obtain an interior Hölder continuity of weak solutions to the nonlocal p-Laplacian type Schrödinger equation by applying the results obtained in Sections 4 and 5. 2. Preliminaries Let Ω ⊂ Rn be a bounded domain with Lipschitz boundary and let K ∈ Kp for p > 1. For p > 1 and 0 < s < 1, let Xs,p(Ω) be the linear function space of all Lebesgue measurable functions v ∈ Fn such that v|Ω ∈ Lp(Ω) and∫∫ R2n Ω |v(x)− v(y)|p |x− y|n+ps dx dy < ∞ where R2n S := R2n\(Sc × Sc) for a set S ⊂ Rn. We also set Xs,p 0 (Ω) = {v ∈ Xs,p(Ω) : v = 0 a.e. in Rn\Ω} (2.1) Since C2 0 (Ω) ⊂ Xs,p 0 (Ω), we see that Xs,p(Ω) and Xs,p 0 (Ω) are nonempty. Then we see that (Xs,p(Ω), ∥ · ∥Xs,p(Ω)) is a normed space with the norm ∥ · ∥Xs,p(Ω) given by ∥v∥Xs,p(Ω) = ∥v∥Lp(Ω) + (∫∫ R2n Ω |v(x)− v(y)|p |x− y|n+ps dx dy )1/p < ∞ (2.2) for v ∈ Xs,p(Ω). For p ≥ 1, we denote by W s,p(Ω) the usual fractional Sobolev space with the norm ∥v∥W s,p(Ω) := ∥v∥Lp(Ω) + [v]W s,p(Ω) < ∞ (2.3) with the seminorm [v]W s,p(Ω) = (∫∫ Ω×Ω |v(x)− v(y)|p |x− y|n+ps dx dy )1/p . When Ω = Rn in (2.3), similarly we define the spaces W s,p(Rn) for p ≥ 1 and s ∈ (0, 1). If p ≥ 1 and s ∈ (0, 1) satisfy ps < n, then it is well-known [11] that there exists a constant c = c(n, p, s,Ω) > 0 such that ∥f∥Lτ (Ω) ≤ c ∥f∥W s,p(Ω) (2.4) for all f ∈ W s,p(Ω) and τ ∈ [p, p∗], where p∗ is the Sobolev exponent p∗ = pn n− ps . Moreover, there is a constant c = c(n, p, s) > 0 such that ∥f∥Lτ (Rn) ≤ c∥f∥W s,p(Rn), ∀τ ∈ [p, p∗], ∥f∥Lp∗ (Rn) ≤ c [f ]W s,p(Rn) ∀f ∈ W s,p(Rn). (2.5) Using (2.5), we easily see that there exists a constant c > 1 depending only on n, p, s and Ω such that ∥u∥Xs,p 0 (Ω) ≤ ∥u∥Xs,p(Ω) ≤ c ∥u∥Xs,p 0 (Ω) ∀u ∈ Xs,p 0 (Ω), (2.6) where ∥u∥Xs,p 0 (Ω) := (∫∫ R2n Ω |u(x)− u(y)|p |x− y|n+ps dx dy )1/p . (2.7) Thus ∥ · ∥Xs,p 0 (Ω) is a norm on Xs,p 0 (Ω) equivalent to (2.2). By using the change of variables, we can easily derive the following version of the fractional Sobolev inequality (2.4). Proposition 2.1. Let BR be a ball with radius R > 0. If s ∈ (0, 1) and p ∈ [1,∞) with sp < n, then there is a constant c = c(n, p, s) > 0 such that ∥f∥Lτ (BR) ≤ cR−n( 1 p− 1 τ )∥f∥Lp(BR) + cR−n( 1 p− 1 τ )+s[f ]W s,p(BR) ∀τ ∈ [p, p∗]. In particular, if τ = p∗ := pn n−ps , we have ∥f∥Lp∗ (BR) ≤ cR−s∥f∥Lp(BR) + c [f ]W s,p(BR). (2.8) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 7 For g ∈ W s,p(Rn), we consider the convex subsets of Xs,p(Ω) defined by Xs,p g (Ω)± = {v ∈ Xs,p(Ω) : (g − v)± ∈ Xs,p 0 (Ω)}, Xs,p g (Ω) := Xs,p g (Ω)+ ∩Xs,p g (Ω)− = {v ∈ Xs,p(Ω) : g − v ∈ Xs,p 0 (Ω)}. For g ∈ W s,p(Rn) and a potential V ∈ Ps,p q (Rn) with q > n ps > 1 (p > 1, 0 < s < 1), let Y s,p(Ω) = Xs,p(Ω) ∩ Lp V (Ω) and Y s,p g (Ω) = Xs,p g (Ω) ∩ Lp V (Ω) where Lp V (Ω) is the weighted Lp class of all real-valued measurable functions u on Rn satisfying −∞ < ∥u∥p Lp V (Ω) := ∫ Ω |u(y)|p V+(y) dy − ∫ Ω |u(y)|p V−(y) dy := ∥u∥p Lp V+ (Ω) − ∥u∥p Lp V− (Ω) < ∞. That is, we see that u ∈ Lp V (Ω) if and only if u ∈ Lp V+ (Ω)∩Lp V− (Ω). Here, we note that ∥u∥p Lp V (Ω) need not be nonnegative, and so the class Lp V (Ω) is not always a normed space. Also we consider function spaces Y s,p g (Ω)+ and Y s,p g (Ω)− defined by Y s,p g (Ω)± = {u ∈ Y s,p(Ω) : (g − u)± ∈ Y s,p 0 (Ω)}. Then we see that Y s,p g (Ω) = Y s,p g (Ω)+ ∩ Y s,p g (Ω)−. If u = g = 0 in Rn\Ω, then we easily know that Y s,p 0 (Ω) = Xs,p 0 (Ω)∩Lp V (Ω) need not be a Banach space. However, if V ∈ Ps,p q (Rn) for q > n ps > 1 (p > 1, 0 < s < 1), then it turns out in Lemma 3.2 below that the class Y s,p 0 (Ω) is a quasi-Banach space with the quasinorm ∥ · ∥Y s,p 0 (Ω) given by ∥u∥p Y s,p 0 (Ω) := ∥u∥p Xs,p 0 (Ω) + ∫ Ω |u(y)|pV (y) dy, u ∈ Y s,p 0 (Ω), Y s,p 0 (Ω) = Xs,p 0 (Ω) and they are quasinorm-equivalent. To define weak solutions of the nonlocal equation (1.3), we consider a bilinear form ⟨·, ·⟩Hp,K : Xs,p(Ω)×Xs,p(Ω) → R defined by ⟨u, v⟩Hp,K = ∫∫ Rn×Rn Hp(u(x)− u(y))(v(x)− v(y)) dK(x, y) where dK(x, y) := K(x− y) dx dy. Definition 2.2. Let V ∈ Ps,p q (Rn) and g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, 0 < s < 1). Then we say that a function u ∈ Y s,p g (Ω)− (u ∈ Y s,p g (Ω)+) is a weak subsolution (weak supersolution) of the nonlocal p-Laplacian type Schrödinger equation (1.3), if it satisfies ⟨u, φ⟩Hp,K + ∫ Rn V (x)|u(x)|p−2u(x)φ(x) dx ≤ 0 (≥ 0) (2.9) for all nonnegative φ ∈ Y s,p 0 (Ω). Also, we say that a function u is a weak solution of the nonlocal equation (1.3), if it is both a weak subsolution and a weak supersolution, i.e. ⟨u, φ⟩Hp,K + ∫ Rn V (x)|u(x)|p−2u(x)φ(x) dx = 0 ∀φ ∈ Y s,p 0 (Ω). (2.10) To prove our results, we need a well-known lemma [15] that is useful in applying the De Giorgi- Nash-Moser method. Lemma 2.3. Let {Nk}∞k=0 ⊂ R be a sequence of positive numbers such that Nk+1 ≤ d0 e k 0N 1+η k ∀k ∈ N ∪ {0}, where d0, η > 0 and e0 > 1. If N0 ≤ d −1/η 0 e −1/η2 0 , then we have Nk ≤ e −k/η 0 N0 for any k = 0, 1, . . . and moreover limk→∞ Nk = 0. We need several elementary inequalities which are useful in proving Theorems 1.2 and 1.5. 8 Y.-C. KIM EJDE-2025/83 Lemma 2.4 ([24]). (a) If a, b ∈ R and A,B ≥ 0, then we have the inequality |b− a|p−2(b− a)(bBp − aAp) ≥ −cp ( |a|+ |b| )p |B −A|p for all p > 1, where cp = (p−1)p−1 2 . (b) If a, b ∈ R with b ≥ a and A,B ≥ 0, then we have the inequality (b− a)p−1(bBp − aAp) ≥ 1 4 (b− a)p(Ap +Bp)− dp ( |a|+ |b| )p |B −A|p for all p > 1, where dp = 1 2 [2(p− 1)]p−1. (c) If A ≥ B ≥ 0 and p > 1, then (A − B)p−1 ≥ bpA p−1 − Bp−1 for p > 1, where bp = 1(1,2](p) + 2−(p−1)1(2,∞)(p). Lemma 2.5 ([10]). If p ≥ 1, ε ∈ (0, 1] and a, b ≥ 0, then ap ≤ bp + cpε b p + (1 + cpε)ε 1−p|a− b|p, where cp = (p− 1)Γ(1 ∨ (p− 2)) for the standard Gamma function Γ. 3. Weighted norm inequalities on the Ap-Muckenhoupt class In this section, we briefly introduce the Ap-Muckenhoupt class for p ≥ 1 and we prove that any potential V in Ps,p q (Rn) (q > n ps > 1, p > 1, 0 < s < 1) satisfies certain weighted norm inequalities related with (V−, V+). By a weight ω on Rn given by the Lebesgue measure, we mean a locally integrable function ω : Rn → [0,∞) almost everywhere. For f ∈ L1 loc(Rn) and x ∈ Rn, the Hardy-Littlewood maximal function Mf is defined by Mf(x) = sup Qx − ∫ Qx |f(y)| dy, where the supremum is taken over all cubes Qx with center x. For a pair (υ, ω) of weights, the quantity [υ, ω](Ap,Ap) is defined by [υ, ω](Ap,Ap) = { supQ ( − ∫ Q υ(y) dy )( − ∫ Q ω(y)− 1 p−1 dy )p−1 , 1 < p < ∞, supQ ( − ∫ Q υ(y) dy ) ∥ω−1∥L∞(Q), p = 1, where the supremum is taken over all cubes Q ⊂ Rn. For 1 ≤ p < ∞, we say that (υ, ω) ∈ (Ap, Ap) if [υ, ω](Ap,Ap) < ∞, and ω ∈ Ap if [ω]Ap := [ω, ω](Ap,Ap) < ∞. For 1 ≤ p < ∞, the facts that (A1, A1) ⊂ (Ap, Ap), A1 ⊂ Ap and (υ, ω) ∈ (Ap, Ap) is equivalent to the mapping property that M : Lp ω(Rn) → Lp,∞ υ (Rn) is bounded, i.e. there is a universal constant Cn,p > 0 such that sup t>0 [ t υ ( {x ∈ Rn : Mf(x) > t} )1/p] ≤ Cn,p (∫ Rn |f(y)|pω(y) dy )1/p (3.1) for any f ∈ Lp ω(Rn), are well-known in [16]. Here, we denote by υ(E) = ∫ E υ(y) dy for a set E ⊂ Rn. The reader can refer to [16] for these stuffs in Fourier analysis. We shall now furnish several examples in the class Ps,p q (Rn) for q > n ps > 1 (p > 1, 0 < s < 1) mentioned in the above introduction (see also [24]): (a) If gα(x) = |x|α for α ∈ R, then it is easy to check that [gη]A1 < ∞ if and only if − n < α ≤ 0. If we consider a sign-changing potential V1(x) = |x|α/q cos(|x|) with α ∈ (−n, 0] and q > n ps > 1 (p > 1, 0 < s < 1), then we can easily check that V1 ∈ Ps,p q (Rn) with b(x) = cos+(|x|). EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 9 (b) Let ν be a Borel measure on Rn satisfying that Mν(x) := sup Qx ν(Qx) |Qx| ≤ C for a.e. x ∈ Rn, where the supremum is taken over all cubes Qx with center x ∈ Rn. Then it is known in [14] that [hγ ]A1 < ∞ for hγ(x) = [Mν(x)]γ , γ ∈ (0, 1). We consider the following sign-changing potential V2(x) = hγ(x) sin(1/|x|) for γ ∈ (0, 1). If q > n ps > 1 for p > 1 and 0 < s < 1, then it is easy to check that V2 ∈ Ps,p q (Rn) with b(x) = sin+(1/|x|). (c) Let υ(x) = ln(1/|x|)1B(0;e−1) + 1Rn\B(0;e−1). Then it is easy to check that υ ∈ A1. We consider the following sign-changing potential V3(x) = υ(x) cos(1/|x|). If q > n ps > 1 for p > 1 and 0 < s < 1, then it follows from properties of the Gamma function that V3 ∈ Ps,p q (Rn) with b(x) = cos+(1/|x|). Next, we derive several fundamental lemmas which are useful in proving Theorem 3.3. Lemma 3.1. If V ∈ Ps,p q (Rn) for q > n ps > 1, p > 1 and 0 < s < 1, then∫ Rn |φ(y)|p V−(y) dy ≤ ∫ Rn |φ(y)|p V+(y) dy for all φ ∈ Y s,p 0 (Ω). Proof. In the exactly same way as the proof of [23, Theorem 3.6], we see that∫ Rn |φ(y)|p V (y) dy ≥ 0 (3.2) for all φ ∈ C∞ c (Rn). Take any φ ∈ Y s,p 0 (Ω). Since C∞ c (Ω) is dense in Xs,p 0 (Ω) (see [13] and [17, Theorem 1.4.2.2]), we can take a sequence {φk} ⊂ C∞ c (Ω) such that φk → φ in Xs,p 0 (Ω). Since V ∈ Ps,p q (Rn), there is a nonnegative bounded function b on Rn such that V b + := V+ b ∈ Lq loc(R n), (V−, V b,i + ) ∈ (A1, A1) and V b,i + ∈ A1 (3.3) for all i ∈ N. Then we claim that lim k→∞ ∫ Rn |φk(y)|p V+(y) dy = ∫ Rn |φ(y)|p V+(y) dy; (3.4) indeed, we note that ∥φk∥Xs,p 0 (Ω) ≤ 2∥φ∥Xs,p 0 (Ω) for all sufficiently large k ∈ N, and also we see that, for any y ∈ Rn, |φk(y)|p − |φ(y)|p = ∫ |φk(y)| |φ(y)| d dτ τp dτ ≤ p ( |φk(y)| − |φ(y)| ) ( |φk(y)| ∨ |φ(y)| )p−1 . Thus it follows from the fractional Sobolev inequality and Hölder’s inequality that∣∣∣ ∫ Rn |φk(y)|p V+(y) dy − ∫ Rn |φ(y)|p V+(y) dy ∣∣∣ ≤ ∫ Rn ∣∣|φk(y)|p − |φ(y)|p ∣∣V+(y) dy ≤ p (∫ Ω ∣∣|φk(y)| − |φ(y)| ∣∣pV+(y) dy )1/p(∫ Ω ( |φk(y)| ∨ |φ(y)| )p V+(y) dy ) p−1 p ≲ ∥φ∥Xs,p 0 (Ω)∥φk − φ∥Xs,p 0 (Ω)∥V+∥Lq(Ω) → 0 as k → ∞. (3.5) Also, we claim that lim k→∞ ∫ Rn |φk(y)|pV−(y) dy = ∫ Rn |φ(y)|p V−(y) dy; (3.6) 10 Y.-C. KIM EJDE-2025/83 indeed, we have V−(y) ≤ [V−, V b,1 + ](A1,A1)V b,1 + (y) a.e. y ∈ Rn, because (V−, V b,1 + ) ∈ (A1, A1) by (3.3). Since V b + ∈ Lq loc(Rn) by (3.3), as in (3.5), we obtain that∣∣∣ ∫ Rn |φk(y)|p V−(y) dy − ∫ Rn |φ(y)|p V−(y) dy ∣∣∣ ≤ ∫ Rn ∣∣|φk(y)|p − |φ(y)|p ∣∣V−(y) dy ≤ ( ∥b∥L∞(Rn) ∨ 1 ) [V−, V b,1 + ](A1,A1) × ∫ Ω ∣∣|φk(y)|p − |φ(y)|p ∣∣V b +(y) dy → 0 as k → ∞. Thus, by (3.2), (3.4) and (3.6), we obtain that∫ Rn |φ(y)|p V−(y) dy = lim k→∞ ∫ Rn |φk(y)|p V−(y) dy ≤ lim k→∞ ∫ Rn |φk(y)|p V+(y) dy = ∫ Rn |φ(y)|p V+(y) dy. The proof is complete. □ Lemma 3.2. If V ∈ Ps,p q (Rn) for q > n ps > 1 (p > 1, 0 < s < 1), then Y s,p 0 (Ω) is a quasi-Banach space with the quasinorm ∥ · ∥Y s,p 0 (Ω) given by ∥u∥p Y s,p 0 (Ω) := ∥u∥p Xs,p 0 (Ω) + ∫ Ω |u(y)|pV (y) dy, u ∈ Y s,p 0 (Ω). Moreover, Y s,p 0 (Ω) = Xs,p 0 (Ω) and they are quasinorm-equivalent. Proof. It follows from Lemma 3.1 and (2.5) that ∥u∥p Xs,p 0 (Ω) ≤ ∥u∥p Y s,p 0 (Ω) ≤ ∥u∥p Xs,p 0 (Ω) + ∥u∥p Lp V+ (Ω) ≤ (1 + |Ω| ps n − 1 q )∥u∥p Xs,p 0 (Ω) . Since Xs,p 0 (Ω) is a Banach space, this implies the required results. □ Theorem 3.3. If V ∈ Ps,p q (Rn) for q > n ps > 1 and 0 < s < 1, then∫ Rn |φ(y)|V−(y) dy ≤ ∫ Rn |φ(y)|V+(y) dy (3.7) for all φ ∈ Y s,p 0 (Ω). Proof. Take any φ ∈ Y s,p 0 (Ω). Since C∞ c (Ω) is dense in Xs,p 0 (Ω) (see [17] and [13]), Y s,p 0 (Ω) = Xs,p 0 (Ω) and they are quasinorm-equivalent by Lemma 3.2, we can take a sequence {φi} ⊂ C∞ c (Ω) such that φi → φ in Y s,p 0 (Ω). So by (2.5) we also have φi → φ in Lp∗(Ω), where p∗ = pn n−ps . So we can choose a subsequence {φik} such that φik → φ a..e. in Ω. (3.8) Also we have lim k→∞ ∫ Rn |φik(y)|V+(y) dy = ∫ Rn |φ(y)|V+(y) dy; (3.9) indeed, it follows from the fractional Sobolev inequality and Hölder’s inequality that∣∣∣ ∫ Rn |φik(y)|V+(y) dy − ∫ Rn |φ(y)|V+(y) dy ∣∣∣ ≤ ∫ Ω ∣∣|φik(y)| − |φ(y)| ∣∣V+(y) dy ≤ (∫ Ω ∣∣|φik(y)| − |φ(y)| ∣∣p V+(y) dy )1/p(∫ Ω V+(y) dy ) p−1 p EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 11 ≲ |Ω| s n+ 1 q′ − 1 p ∥φik − φ∥Y s,p 0 (Ω)∥V+∥Lq(Ω) → 0 as k → ∞ where q′ is the dual exponent of q. Hence, by Fatou’s lemma, Lemma 3.1, [23, Lemma 3.5], (3.2), (3.4), (3.8) and (3.9), we conclude that∫ Rn |φ(y)|V−(y) dy = lim p→1+ ∫ Rn |φ(y)|p V−(y) dy ≤ lim p→1+ lim inf k→∞ ∫ Rn |φik(y)|p V−(y) dy ≤ lim p→1+ lim inf k→∞ ∫ Rn |φik(y)|p V+(y) dy ≤ lim p→1+ ∫ Rn |φ(y)|p V+(y) dy = ∫ Rn |φ(y)|V+(y) dy. Therefore the proof is complete. □ 4. Local properties of weak subsolutions In this section, we shall obtain certain local properties for weak subsolutions to the nonlocal p-Laplacian type Schrödinger equation. These results play a crucial role in establishing an interior Hölder continuity for weak solutions to the nonlocal equation (1.3). To establish the result, we need several steps. Lemma 4.1. For p > 1 and N > 0, let h(t) = tp−1(t−N)+ − (t−N)p+ ≥ 0. Then h is Lipschitz continuous on R. Proof. We note that h(t) = 0 for t < N , h is in C1(N,∞), lim t→N+ h(t)− h(N) t−N = 0 and lim t→N− h(t)− h(N) t−N = Np−1. Also it is easy to check that limt→∞ h(t) = 0, because lim t→∞ tp−1(t−N)+ (t−N)p+ = 1. Moreover, in order to check the differentiability of h at the infinity, we set g(t) = h(1/t). Then we have lim t→0+ g(t)− g(0) t = lim t→0+ [ 1 tp (1 t −N ) − 1 t (1 t −N )p] = 0, because lim t→0+ [ 1 tp (1 t −N )/1 t (1 t −N )p] = lim t→0+ 1− tN (1− tN)p = 1. This implies the required result. □ Corollary 4.2. If N > 0 and u ∈ Xs,p(Ω) for p > 1 and 0 < s < 1, then (a) |u|p−2u(u−N)+ − (u−N)p+ ∈ Xs,p(Ω) and it is nonnegative in Rn, and (b) [ |u|p−2u(u−N)+ − (u−N)p+ ] ζp ∈ Xs,p 0 (Ω) for any nonnegative ζ ∈ C∞ c (Ω). Proof. (a) Note that |u|p−2u(u−N)+ − (u−N)p+ = h ◦ u for the function h given in Lemma 4.1. Since h is Lipschitz continuous on R by Lemma 4.1, it is obvious that 0 ≤ h ◦ u ∈ Xs,p(Ω). (b) By the mean value theorem, we have |ζp(x)− ζp(y)| = ∣∣ ∫ ζ(x) ζ(y) d dτ τp dτ ∣∣ ≤ p ( ζp−1(x) ∨ ζp−1(y) ) |ζ(x)− ζ(y)| ≤ 2p∥ζ∥p−1 L∞(Rn)|ζ(x)− ζ(y)| 12 Y.-C. KIM EJDE-2025/83 for all x, y ∈ Rn. This means that ζp is Lipschitz continuous on Rn. So we can easily derive the result. The proof is complete. □ Next we need the nonlocal Caccioppoli type inequality that is Theorem 1.5. This is a very useful tool in proving local boundedness of weak supersolutions to the nonlocal p-Laplacian type Schrödinger equation. Proof of Theorem 1.5. For simplicity, we assume that x0 = 0. Let w = (u−M)+ for M ∈ [0,∞) and take any nonnegative ζ ∈ C∞ c (Br). We use φ = wζp as a testing function in the weak formulation of the equation. Then we have ⟨u, φ⟩Hp,K + ∫ Rn |u(y)|p−2u(y)φ(y)V (y) dy ≤ 0, (4.1) where dK(x, y) = K(x− y) dx dy and ⟨u, φ⟩Hp,K = ∫∫ Rn×Rn Hp(u(x)− u(y))(φ(x)− φ(y)) dK(x, y) for p ≥ 1. The first term in the left-hand side of the above inequality can be decomposed into two parts as follows: ⟨u, φ⟩Hp,K = ∫∫ Br×Br Hp(u(x)− u(y))(φ(x)− φ(y)) dK(x, y) + 2 ∫∫ (Rn\Br)×Br Hp(u(x)− u(y))φ(x) dK(x, y) := I1 + 2I2. (4.2) For estimating I1, without loss of generality we assume that u(x) ≥ u(y). Then we first observe that w(x) ≥ w(y) and Hp(u(x)− u(y))(φ(x)− φ(y)) ≥ (w(x)− w(y))p−1(φ(x)− φ(y)) (4.3) whenever x, y ∈ Br; indeed, it can easily be checked by considering three possible occasions (i) u(x), u(y) > M , (ii) u(x) > M , u(y) ≤ M , and (iii) u(y) ≤ u(x) ≤ M . Also we observe that |ζ(x)w(x)− ζ(y)w(y)|p ≤ 2p−1|w(x)− w(y)|p(ζp(x) + ζp(y)) + 2p−1(wp(x) + wp(y))|ζ(y)− ζ(x)|p. (4.4) By (b) of Lemma 2.4, (4.3) and (4.4), we have Hp(u(x)− u(y))(φ(x)− φ(y)) ≥ 1 4 |w(x)− w(y)|p(ζp(x) + ζp(y))− dp(w p(x) + wp(y))|ζ(y)− ζ(x)|p ≥ 1 2p+1 |ζ(x)w(x)− ζ(y)w(y)|p − (1 4 + dp ) (wp(x) + wp(y))|ζ(y)− ζ(x)|p. (4.5) Thus it follows that I1 ≥ 1 2p+1 ∫∫ Br×Br |ζ(x)w(x)− ζ(y)w(y)|p dK(x, y) − 2p (1 4 + dp ) ∫∫ Br×Br (wp(x) + wp(y))|ζ(y)− ζ(x)|p dK(x, y). (4.6) For the estimate of I2, we note that Hp(u(x)− u(y))φ(x) ≥ −(u(y)− u(x))p−1 + (u(x)−M)+ζ p(x) ≥ −(u(y)−M)p−1 + (u(x)−M)+ζ p(x) = −wp−1(y)w(x)ζp(x) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 13 and thus we have I2 ≥ − ∫∫ (Rn\Br)×Br wp−1(y)w(x)ζp(x) dK(x, y) ≥ − ( sup x∈supp(ζ) ∫ Rn\Br wp−1(y)K(x− y) dy )∫ Br w(x)ζp(x) dx. (4.7) Finally, we claim that∫ Rn |u(y)|p−2u(y)φ(y)V (y) dy ≥ ∫ Rn wp(y)ζp(y)V (y) dy. (4.8) This is equivalent to the inequality∫ Rn [ |u(y)|p−2u(y)w(y)− wp(y) ] ζp(y)V+(y) dy ≥ ∫ Rn [ |u(y)|p−2u(y)w(y)− wp(y) ] ζp(y)V−(y) dy, whose proof is just a direct application of Lemma 3.2, Theorem 3.3 and Corollary 4.2. Hence the required inequality can be obtained from (4.1), (4.2), (4.6), (4.7) and (4.8). □ Next, we shall obtain the local boundedness of such weak subsolutions which is Theorem 1.1 and a relation between the nonlocal tail terms of the positive part and the negative part of weak solutions of the nonlocal p-Laplacian type Schrödinger equation (1.3) in the following theorems. Proof of Theorem 1.1. Take any ζ ∈ C∞ c (B0 r ) such that |∇ζ| ≤ c/r on Rn. Let w = (u−M)+ for M ∈ (0,∞). By Lemma 3.2, Theorem 1.5 and the mean value theorem, we have∫∫ B0 r×B0 r |ζ(x)w(x)− ζ(y)w(y)|p dK(x, y) ≲ rp−ps∥∇ζ∥pL∞(B0 r) ∥w∥pLp(B0 r) +A(w, ζ, r, s) ∥w∥L1(B0 r) (4.9) where A(w, ζ, r, s) = sup x∈supp(ζ) ∫ Rn\B0 r wp−1(y)K(x− y) dy. Applying Proposition 2.1 to (4.9), we obtain( − ∫ B0 r |wζ|pγ dx )1/γ ≲ ( rp−ps∥∇ζ∥pL∞(B0 r) + r−ps ) rps− ∫ B0 r |w|p dx+A(w, ζ, r, s) rps− ∫ B0 r w dx (4.10) where γ = n n−ps > 1. For k = 0, 1, 2, . . . , we set rk = (1 + 2−k)r, r∗k = rk + rk+1 2 , Mk = M + (1− 2−k)M∗, M∗ k = Mk +Mk+1 2 , wk = (u−Mk)+ and w∗ k = (u−M∗ k )+ for a constant M∗ > 0 to be determined later. In (4.9), for k = 0, 1, . . . , we choose a function ζk ∈ C∞ c (B0 r∗k ) with ζk|B0 rk+1 ≡ 1 such that 0 ≤ ζk ≤ 1 and |∇ζk| ≤ c 2k+2/r in Rn. For k = 0, 1, 2, . . . , we set Nk = ( − ∫ B0 rk |wk|p dx )1/p . Since w∗ k ≥ wk+1 and w∗ k(x) ≥ Mk+1 −M∗ k = 2−k−2M∗ whenever u(x) ≥ Mk+1, we then have Nk+1 ≲ ( 1 |B0 rk | ∫ B0 rk+1 wp k+1(w ∗ k) p(γ−1) (Mk+1 −M∗ k ) p(γ−1) dx )1/p ≲ ( 2k M∗ )γ−1( − ∫ B0 rk |w∗ k ζk|pγ dx )1/p . (4.11) 14 Y.-C. KIM EJDE-2025/83 Since ζk ∈ C∞ c (B0 rk ), w∗ k ≤ w0 for all k and |y − x| ≥ |y − x0| − |x− x0| ≥ ( 1− r∗k rk ) |y − x0| ≥ 2−k−2|y − x0| for any x ∈ B0 r∗k and y ∈ Rn\B0 rk , we easily obtain that A(w∗ k, ζk, rk, s) ≤ c 2k(n+ps) r−ps [Tr(w0;x0)] p−1. (4.12) Since 0 ≤ w∗ k ≤ wk and wk(x) ≥ M∗ k −Mk = 2−k−2M∗ if u(x) ≥ M∗ k , it follows from (4.10)–(4.12) that ( 2k M∗ )− p(γ−1) γ N p γ k+1 ≤ c2pk − ∫ B0 rk |w∗ k|p dx+ c2k(n+ps) (rk r )ps [Tr(w0;x0)] p−1 − ∫ B0 rk w∗ k dx ≤ c2pkNp k + c[Tr(w0;x0)] p−1 2k(n+ps) − ∫ B0 rk w∗ kw p−1 k (M∗ k −Mk)p−1 dx ≤ c ( 2pk + 2k(n+ps) ( 2k M∗ )p−1 [Tr(w0;x0)] p−1 ) Np k . Taking M∗ in the above so that M∗ ≥ δTr(w0;x0) for δ ∈ (0, 1], we obtain that Nk+1 M∗ ≤ d0 a k (Nk M∗ )1+η where d0 = c γ p δ− p−1 p γ > 0, a = 2 γ p (n+2s+p−1)+ ps n−ps > 1 and η = γ − 1 > 0. If N0 ≤ d − 1 η 0 a − 1 η2 M∗, then we set M∗ = δ Tr(w0;x0) + c0 δ − (p−1)n sp2 a (n−ps)2 p2s2 N0 where c0 = c n sp2 . By Lemma 2.3, we conclude that sup B0 r u ≤ M +M∗ ≤ M + δTr(w0;x0) + c0δ − (p−1)n sp2 a (n−ps)2 p2s2 ( − ∫ B0 2r (u−M)p+ dx )1/p . Hence, taking M ↓ 0 in the above estimate, we obtain the required result. □ The third estimate is a lemma which furnishes a relation between the nonlocal tails of the positive and negative part of weak solutions to the nonlocal p-Laplacian type Schrödinger equation. Lemma 4.3. Let V ∈ Ps,p q (Rn), g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, s ∈ (0, 1)). If u ∈ Y s,p g (Ω) is a weak solution of the nonlocal p-Laplacian type Schrödinger equation (1.3) such that u ≥ 0 in B0 R ⊂ Ω, then we have the estimate Tr(u+;x0) ≲ ( 1 + ∥V+∥Lq(Ω) ) sup B0 r u+ ( r R ) ps p−1 TR(u−;x0) ∀r ∈ (0, R). Proof. Without loss of generality, we assume that x0 = 0. Let M = supBr u and φ(x) = w(x)ζp(x) where w(x) = u(x) − 2M and ζ ∈ C∞ c (B3r/4) is a function satisfying that ζ|Br/2 ≡ 1, 0 ≤ ζ ≤ 1 and |∇ζ| ≤ c/r in Rn. Then we have 0 = ∫∫ Br×Br Hp(u(x)− u(y))(φ(x)− φ(y)) dK(x, y) + 2 ∫ Rn\Br ∫ Br Hp(u(x)− u(y))(u(x)− 2M)ζp(x) dK(x, y) + ∫ Rn V (x)|u(x)|p−2u(x)(u(x)− 2M)ζp(x) dx := J1 + J2 + J3. (4.13) Since −2M ≤ w(x) := u(x)− 2M ≤ −M ∀x ∈ Br, (4.14) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 15 by Lemma 2.4(a) we have Hp(w(x)− w(y))(w(x)ζp(x)− w(y)ζp(y)) ≥ −cp4 pMp(ζ(x)− ζ(y))p for any x, y ∈ Br, it follows from simple calculation that J1 ≥ −cp4 pMp ∫∫ Br×Br (ζ(x)− ζ(y))p dK(x, y) ≳ −Mpr−ps|Br|. (4.15) The lower estimate on J2 can be split as follows J2 ≥ 4 ∫ Rn\Br ∫ Br M(u(y)−M)p−1 + ζp(x) dK(x, y) − 4M ∫ EM ∫ Br (u(x)− u(y))p−1 + ζp(x) dK(x, y) := J2,1 − J2,2, where EM = {y ∈ Rn\Br : u(y) < M}. Since (u(y) −M)+ ≥ u+(y) −M , it follows from (c) of Lemma 2.4 that (u(y)−M)p−1 + ≥ bpu p−1 + (y)−Mp−1 where bp = 1(1,2](p) + 2−(p−1)1(2,∞)(p). Thus the lower estimate on J2,1 can be obtained as J2,1 ≥ d2Mr−ps|Br| [ Tr(u+; 0) ]p−1 − d3M pr−ps|Br| (4.16) with universal constants d2, d3 > 0. If x ∈ Br and y ∈ EM , then we observe that (u(x)− u(y))p−1 + ≤ ap ( |u(x)−M |p−1 + |M − u(y)|p−1 ) ≤ apM p−1 + ap(M + u−(y)− u+(y)) p−1 ≤ apM p−1 + ap(M + u−(y)) p−1 ≤ ap(1 + ap)M p−1 + a2p [u−(y)] p−1 where ap = 1(1,2](p)+2p−11(2,∞)(p), because u+(y) < M+u−(y) for any y ∈ EM . Since u−(y) = 0 for all y ∈ BR, the upper estimate on J2,2 can thus be achieved by J2,2 ≤ 4ap(1 + ap)M p ∫ Rn\Br ∫ Br ζp(x) dK(x, y) + 4apM ∫ Rn\BR ∫ Br [u−(y)] p−1ζ2(x) dK(x, y) ≤ d4M pr−ps|Br|+ d5MR−ps|Br| [ TR(u−; 0) ]p−1 (4.17) with universal constants d4, d5 > 0. Thus, by (4.16) and (4.17), we have J2 ≥ −dMpr−ps|Br| − dMR−ps|Br| [ TR(u−; 0) ]p−1 + eMr−ps|Br| [ Tr(u+; 0) ]p−1 (4.18) where d, e > 0 are some universal constants depending only on n, s, λ and Λ. Finally, it follows from (4.14), Hölder’s inequality and the fractional Sobolev inequality that J3 ≥ −2Mp ∫ Rn V+(x)ζ p(x) dx ≥ −2Mp∥V+∥Lq(Ω) (∫ Ω ζpq ′ (x) dx )1/q′ ≥ −2Mp∥V+∥Lq(Ω) (∫ Ω ζ pn n−ps (x) dx )n−ps n |Ω| 1 q′ − n−ps n ≥ −2Mp∥V+∥Lq(Ω)|Ω| 1 q′ − n−ps n ( r−ps∥ζ∥pLp(Br) + [ζ]pW s,p(Br) ) ≳ −∥V+∥Lq(Ω)M pr−ps|Br| (4.19) where q > n ps > 1 and 1 < q′ < n n−ps with 1 q + 1 q′ = 1. Hence the estimates (4.13), (4.15), (4.18), and (4.19) give the required estimate. □ 16 Y.-C. KIM EJDE-2025/83 Next we shall obtain the local boundedness for nonnegative weak solutions of the nonlocal equation (1.3) by employing Theorem 1.1 and Lemma 4.3. It is interesting that this estimate no longer depends on the nonlocal tail term, whose proof is pretty simple. Theorem 4.4. Let V ∈ Ps,p q (Rn) and g ∈ W s,p(Rn) for q > n ps > 1 (p > 1, s ∈ (0, 1)). If u ∈ Y s,p g (Ω) is a nonnegative weak solution of the nonlocal p-Laplace type Schrödinger equation (1.3), then we have the estimate sup B0 r u ≤ C ( − ∫ B0 2r up(x) dx )1/p for any r > 0 with B0 2r ⊂ Ω. Proof. We choose some δ ∈ (0, 1] so that 1− δd0 > 0 and take any r > 0 with B0 2r ⊂ Ω where d0 = c0(1 + ∥V+∥Lq(Ω)) > 0 for the universal constant c0 > 0 given in Lemma 4.3. Then it follows from Theorem 1.1 and Lemma 4.3 that sup B0 r u ≤ δ d0 [ sup B0 r u+ T2r(u−;x0)) ] + C0 δ − (p−1)n sp2 ( − ∫ B0 2r up(x) dx )1/p Since T2r(u−;x0) = 0, we can easily derive the required result by taking C = C0 δ − (p−1)n sp2 1− δd0 . Hence we complete the proof. □ 5. Logarithm of a weak solution is a locally bounded mean oscillation function In this section, we prove that the logarithm of a weak supersolution to the nonlocal p-Laplacian type Schrödinger equation (1.3) is a function with locally bounded mean oscillation. To do this, the following tool which is called the fractional Poincaré inequality is very useful. Let n ≥ 1, p ≥ 1, s ∈ (0, 1) and sp < n. For a ball B ⊂ Rn, let uB denote the average of u ∈ W s,p(B) over B, i.e. uB = − ∫ B u(y) dy. Then it was shown in [2, 31] that ∥u− uB∥pLp(B) ≤ cn,p(1− s)|B| sp n (n− sp)p−1 [u]pW s,p(B) (5.1) with a universal constant cn,p > 0 depending only on n and p, which is usually very useful in getting the logarithmic estimate of weak supersolutions. Of course, the logarithmic estimate could be obtained as in [10], but we will not apply their approach to achieve it. Our method to realize the logarithmic estimate is easier and more simple than their method. Proof of Theorem 1.2. For simplicity, we set x0 = 0. So, in what follows, we write Br := B0 r for r > 0. Take any r > 0 so that B2r ⊂ BR where BR ⊂ Ω. Consider a radial function ζ ∈ C∞ c (B3r/2) with values in [0, 1] such that ζ|Br ≡ 1, ζ|Rn\B2r ≡ 0 and |∇ζ| ≲ 1 r in Rn. We use the function φ(x) = ζp(x) up−1 b (x) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 17 as a testing function to the nonlocal p-Laplacian type Schrödinger equation (1.3), where ub(x) = u(x) + b. Then we have 0 ≤ ∫∫ Rn×Rn Hp(ub(x)− ub(y))(φ(x)− φ(y)) dK(x, y) + ∫ Rn V (x)Hp(u(x))φ(x) dx = ∫∫ B2r×B2r Hp(ub(x)− ub(y))(φ(x)− φ(y)) dK(x, y) + 2 ∫ Rn\B2r ∫ B2r Hp(ub(x)− ub(y))φ(x) dK(x, y) + ∫ Rn V (x)|u(x)|p−2u(x)φ(x) dx := H(u, φ) + I(u, φ) + J(u, φ). (5.2) Without loss of generality, we may assume that ub(x) ≥ ub(y) for the estimate H(u, φ); for, by symmetry, the other case ub(x) < ub(y) can be treated in the exactly same way. Then we have two possible cases: (a) ub(x) ≤ 2ub(y) and (b) ub(x) > 2ub(y). Case (a): ub(y) ≤ ub(x) ≤ 2ub(y). By the mean value theorem, we note that ζ(x) ≥ ζ(y) ⇒ ζp(x)− ζp(y) = p ∫ ζ(x) ζ(y) τp−1dτ ≤ pζp−1(x)(ζ(x)− ζ(y)), ζ(x) < ζ(y) ⇒ ζp(x)− ζp(y) = p ∫ ζ(y) ζ(x) (−τp−1)dτ ≤ pζp−1(x)(ζ(x)− ζ(y)). (5.3) Then it follows that φ(x)− φ(y) = ζp(x)− ζp(y) up−1 b (y) + ζp(x) ( 1 up−1 b (x) − 1 up−1 b (y) ) ≤ pζp−1(x)(ζ(x)− ζ(y)) up−1 b (y) + ζp(x) ∫ 1 0 d dτ ( 1 [τ(ub(x)− ub(y)) + ub(y)]p−1 ) dτ ≤ pζp−1(x)(ζ(x)− ζ(y)) up−1 b (y) − (p− 1) ζp(x)(ub(x)− ub(y)) up b(x) ≤ pζp−1(x) |ζ(x)− ζ(y)|ub(y) up b(y) − (p− 1) 2p ζp(x)(ub(x)− ub(y)) up b(y) . (5.4) Applying Young’s inequality with indices p′ = p p−1 , p, ε, it follows from (5.2) that H(u, φ) ≤ cn,p,sΛp ∫∫ B2r×B2r ε(ub(x)− ub(y)) pζp(x) + cε|ζ(x)− ζ(y)|pup b(y) up b(y) dx dy |x− y|n+ps − cn,p,sλ(p− 1) 2p ∫∫ B2r×B2r (ub(x)− ub(y)) pζp(x) up b(y) dx dy |x− y|n+ps . (5.5) If we choose ε = λ(p−1) 2p+1pΛ in (5.5), then we have H(u, φ) ≲ − ∫∫ B2r×B2r ζp(x) (ub(x)− ub(y)) p up b(y) dx dy |x− y|n+ps + ∫∫ B2r×B2r |ζ(x)− ζ(y)|p |x− y|n+ps dx dy ≲ − ∫∫ B2r×B2r ζp(x) (ub(x)− ub(y)) p up b(y) dx dy |x− y|n+ps + rn−ps. (5.6) because x, y ∈ B2r. Since 0 ≤ ub(x)− ub(y) ≤ ub(y), we have ∣∣lnub(x)− lnub(y) ∣∣p = (∫ 1 0 ub(x)− ub(y) τ(ub(x)− ub(y)) + ub(y) dτ )p ≤ (ub(x)− ub(y)) p up b(y) . (5.7) 18 Y.-C. KIM EJDE-2025/83 Thus by (5.6) and (5.7) we have H(u, φ) ≲ − ∫∫ B2r×B2r ζp(x) ∣∣∣ ln (ub(x) ub(y) )∣∣∣p dx dy |x− y|n+ps + rn−ps ≲ − ∫∫ Br×Br ∣∣∣ ln(ub(x) ub(y) )∣∣∣p dx dy |x− y|n+ps + rn−ps. (5.8) Case (b): ub(x) > 2ub(y). It follows from the inequality in Lemma 2.5 with ε = (2p−1 − 1)/2 that φ(x)− φ(y) = ζp(x)− ζp(y) up−1 b (x) + ζp(y) ( 1 up−1 b (x) − 1 up−1 b (y) ) ≤ ζp(x)− ζp(y) up−1 b (x) + ζp(y) ( 1 2p−1up−1 b (y) − 1 up−1 b (y) ) ≤ εζp(y) + cε|ζ(x)− ζ(y)|p up−1 b (x) − (1− 2−p+1) ζp(y) up−1 b (y) ≤ c |ζ(x)− ζ(y)|p up−1 b (x) − (1 2 − 1 2p ) ζp(y) up−1 b (y) . (5.9) Since ub(x) ≥ ub(x)− ub(y) ≥ ub(y), by (5.2) and (5.9) we have H(u, φ) cn,p,s ≤ cΛ ∫∫ B2r×B2r |ζ(x)− ζ(y)|p |x− y|n+ps dx dy − cλ (1 2 − 1 2p )∫∫ B2r×B2r ζp(y) (ub(x)− ub(y)) p−1 up−1 b (y) dx dy |x− y|n+ps . (5.10) Since (ln t)p ≤ c (t− 1)p−1 for t > 2, we have ∣∣lnub(x)− lnub(y) ∣∣p ≤ c (ub(x)− ub(y) ub(y) )p−1 = c (ub(x)− ub(y)) p−1 up−1 b (y) . (5.11) Combining (5.10) with (5.11), it follows that H(u, φ) ≲ − ∫∫ B2r×B2r ζp(y) ∣∣∣ ln(ub(x) ub(y) )∣∣∣p dx dy |x− y|n+ps + ∫∫ B2r×B2r |ζ(x)− ζ(y)|p |x− y|n+ps dx dy ≲ − ∫∫ Br×Br ∣∣∣ ln(ub(x) ub(y) )∣∣∣p dx dy |x− y|n+ps + rn−ps. (5.12) Hence, by (5.8) and (5.12), we conclude that H(u, φ) ≲ − ∫∫ Br×Br ∣∣∣ ln(ub(x) ub(y) )∣∣∣p dx dy |x− y|n+ps + rn−ps. (5.13) For the estimate of I(u, φ), we note that (i) u(y) ≥ 0 and u(x) − u(y) ≤ u(x) for (x, y) ∈ B2r × (BR\B2r) and (ii) (u(x) − u(y))+ ≤ u(x) + u−(y) for (x, y) ∈ B2r × (Rn\BR). Since ζ is supported in B3r/2, the above observations (i) and (ii) yield that I(u, φ) ≤ 2 cn,p,sΛ ∫ B3r/2 ∫ Rn\B2r 1 |y − x0|n+ps dy dx + 2 cn,p,sΛ ∫ B3r/2 ∫ Rn\BR up−1 − (y) up−1 b (x) dy dx |y − x0|n+ps ≲ rn−ps + rn−ps bp−1 ( r R )ps [TR(u−;x0)] p−1 (5.14) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 19 because |y−x| ≥ |y−x0| − |x−x0| ≥ |y−x0|/4 for all (x, y) ∈ B3r/2 × (Rn\B2r). Also, it follows from Hölder’s inequality and Proposition 2.1 that J(u, φ) ≤ ∫ Rn V+(x)ζ p(x) dx ≤ ∥V+∥Lτ (Ω) (∫ Ω ζpτ ′ (x) dx ) 1 τ′ ≤ ∥V+∥Lq(Ω) (∫ Ω ζ pn n−ps (x) dx )n−ps n |Ω| 1 q′ − n−ps n ≤ ∥V+∥Lq(Ω)|Ω| 1 q′ − n−ps n ( r−ps∥ζ∥pLp(Br) + [ζ]pW s,p(Br) ) ≤ ∥V+∥Lq(Ω)|Ω| 1 q′ − n−ps n rn−ps ≲ ∥V+∥Lq(Ω) r n−ps (5.15) where q > n ps > 1, p > 1 and 1 < q′ < n n−ps with 1 q + 1 q′ = 1. By (5.13), (5.14) and (5.15), we obtain that∫∫ Br×Br ∣∣∣ ln(u(x) + b u(y) + b )∣∣∣p dx dy |x− y|n+ps ≲ rn−ps bp−1 ( r R )ps [TR(u−;x0)] p−1 + rn−ps ( 1 + ∥V+∥Lq(Ω) ) for all b ∈ (0, 1) and r ∈ (0, R/2), since x, y ∈ B2r. Hence we complete the proof by applying (1.1). □ We now introduce a sort of local BMO spaces on B0 R ⊂ Ω, i.e. BMOp(B0 R) for p > 0. The norm ∥ · ∥BMOp(B0 R) is defined by ∥f∥BMOp(B0 R) = sup r∈(0,R/2) ( − ∫ B0 r ∣∣f(y)− fB0 r ∣∣p dy)1/p and the space is given by BMOp(B0 R) = {f ∈ L1 loc(Rn) : ∥f∥BMOp(B0 R) < ∞}. If p = 1, we write BMOp(B0 R) = BMO(B0 R). Then we easily see that ∥ |f | ∥BMO(B0 R) ≤ ∥f∥BMO(B0 R) (5.16) because ∣∣|f | − |f |B0 r ∣∣ ≤ |f − fB0 r | and ∥f ± g∥BMO(B0 R) ≤ ∥f∥BMO(B0 R) + ∥g∥BMO(B0 R). (5.17) We observe that a ∧ b = a+ b− |a− b| 2 and a ∨ b = a+ b+ |a− b| 2 for any a, b ∈ R. This implies that ∥f ∨ g∥BMO(B0 R) ≤ ∥f∥BMO(B0 R) + ∥g∥BMO(B0 R), ∥f ∧ g∥BMO(B0 R) ≤ ∥f∥BMO(B0 R) + ∥g∥BMO(B0 R). (5.18) In addition, we can obtain the following John-Nirenberg inequality (as in [16]) by using the Calderón-Zygmund decomposition in harmonic analysis as follows; there exists some constants b1, b2 > 0 depending only on the dimension n such that∣∣{x ∈ B0 r : |f(x)− fB0 r | > λ} ∣∣ ≤ b1e −(b2/∥f∥BMO(B0 R ) )λ|B0 r | for any f ∈ BMO(B0 r ), every r > 0 with B0 2r ⊂ B0 R and B0 R ⊂ Ω, and every λ > 0. By standard analysis, this inequality makes it possible to easily show that If f ∈ BMO(B0 R) for B 0 R ⊂ Ω and 1 < p < ∞, then ∥ · ∥BMO(B0 R) is norm-equivalent to ∥ · ∥BMOp(B0 r) . (5.19) 20 Y.-C. KIM EJDE-2025/83 Lemma 5.1. If we set v(x) = ln ( a+ b u(x) + b ) for a, b ∈ (0, 1), where the function u satisfies the same assumption as Theorem 1.2, then we have − ∫ B0 r |v(x)− vB0 r |p dx ≲ Rb,r,R(u−;x0) for any r ∈ (0, R/2), where Rb,r,R(u−;x0) = 1 bp−1 ( r R )ps [TR(u−;x0)] p−1 + ( 1 + ∥V+∥Lq(Ω) ) . It follows from this that v ∈ BMO(B0 R), and moreover ∥v∥BMO(B0 R) ≤ [Rb,r,R(u−;x0)] 1/p < ∞. Proof. The first part easily follows from the fractional Poincaré inequality (5.1) and Theorem 1.2. Also the second part can be shown by applying the Remark of Theorem 1.1 and Hölder’s inequality because u ∈ W s,p(Rn). □ Corollary 5.2. If we set v̄ = (v ∨ 0) ∧ d for d > 0 with the same v as in Lemma 5.1, then − ∫ B0 r |v̄(x)− v̄B0 r |p dx ≲ Rb,r,R(u−;x0) ∀r ∈ (0, R/2), where Rb,r,R(u−;x0) = 1 bp−1 ( r R )ps [TR(u−;x0)] p−1 + ( 1 + ∥V+∥Lq(Ω) ) . It follows from this that v̄ ∈ BMO(B0 R), and moreover ∥v̄∥BMO(B0 R) ≤ [Rb,r,R(u−;x0)] 1/p < ∞. Proof. Without loss of generality, assume that x0 = 0. By Lemma 5.1, we have − ∫ Br ∣∣|v(x)| − |v|Br ∣∣p dx ≲ Rb,r,R(u−; 0), because ∣∣|v(x)| − |v|Br ∣∣ ≤ |v(x)− vBr |. Then we can easily derive from (5.17) and (5.19) that − ∫ Br |v̄(x)− v̄Br | dx ≲ [Rb,r,R(u−; 0)] 1/p. Finally, the second part can be done as in Lemma 5.1. □ 6. Interior Hölder regularity In this section, we establish an interior Hölder regularity of weak solutions to the nonlocal p-Laplacian type Schrödinger equation (1.3) by applying the previous results obtained in Sections 4 and 5. Proof of Theorem 1.3. Fix any p > 1 and 0 < s < 1 and take any R > 0 with BR(x0) ⊂ Ω. For simplicity, without loss of generality, we assume that x0 = 0. For any k ∈ N∪{0} and r ∈ (0, R/2), we set rk = δkr 2 for δ ∈ ( 0, (1 4 ) p−1 ps ) , Bk = Brk , B∗ k = B2rk . Let us set Ξ(r0) = 2 Tr/2(u; 0) + 2C0 ( − ∫ Br |u|p dx )1/p where C0 > 1 is the constant given in Theorem 1.1. For k ∈ N ∪ {0}, we set Ξ(rk) = (rk r0 )η Ξ(r0) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 21 where η ∈ (0, ps p−1 ) is some constant to be determined later. For our proof, if we set v = u/Ξ(r0), then we have only to prove that oscBk v ≤ Θ(rk) := Ξ(rk) Ξ(r0) (6.1) for any k ∈ N ∪ {0}. We proceed by using the mathematical induction. By the remark of Theorem 1.1, we see that oscB0 v ≤ Θ(r0). Assume that (6.1) holds for all k ∈ {0, 1, . . . ,m}. Then we will show that (6.1) is still true for m+ 1. For this proof, we consider two possible cases; either∣∣B∗ k+1 ∩ { v ≥ infBk v +Θ(rk)/2 }∣∣ |B∗ k+1| ≥ 1 2 (6.2) or ∣∣B∗ k+1 ∩ { v ≤ infBk v +Θ(rk)/2 }∣∣ |B∗ k+1| ≥ 1 2 . (6.3) If (6.2) holds, then we set vk = v − infBk v, and if (6.3) holds, then we set vk = Θ(rk)− ( v − inf Bk v ) . In these two cases, we see that vk ≥ 0 in Bk and∣∣B∗ k+1 ∩ {vk ≥ Θ(rk)/2} ∣∣ |B∗ k+1| ≥ 1 2 . (6.4) Furthermore, vk is a weak solution satisfying sup Bk |vm| ≤ 2Θ(rk) (6.5) for all k ∈ {0, 1, . . . ,m}. Under the induction hypothesis, if m ≥ 1, then we now claim that [Trk(vk; 0)]p−1 ≤ c δ−(p−1)η [Θ(rk)] p−1 (6.6) for k ∈ {0, 1, . . . ,m}. Indeed, by (6.5) and that∫ Rn\B0 |vm(x)|p−1 |x|n+ps dx ≲ r−ps 0 sup B0 |v|p−1 + r−ps 0 [Θ(r0)] p−1 + ∫ Rn\B0 |v(x)|p−1 |x|n+ps dx ≲ r−ps 1 [Θ(r0)] p−1, we have the estimate [Trm(vm; 0)]p−1 = c rpsm m∑ k=1 ∫ Bk−1\Bk |vm(x)|p−1 |x|n+ps dx+ c rpsm ∫ Rn\B0 |vm(x)|p−1 |x|n+ps dx ≲ rpsm m∑ k=1 [ sup Bk−1 |vm| ]p−1 ∫ Rn\Bk 1 |x|n+ps dx+ rpsm ∫ Rn\B0 |vm(x)|p−1 |x|n+ps dx ≤ m∑ k=1 (rm rk )ps [Θ(rk−1)] p−1 = m∑ k=1 (rm rk )ps(rk−1 r0 )(p−1)η = (rm r0 )(p−1)η m∑ k=1 (rm rk )ps−(p−1)η(rk−1 rk )(p−1)η = [Θ(rm)]p−1δ−(p−1)η m∑ k=1 δ(m−k)[ps−(p−1)η] 22 Y.-C. KIM EJDE-2025/83 ≤ δ−(p−1)η δ (m−1)[ps−(p−1)η] 1− δ−[ps−(p−1)η] [Θ(rm)]p−1 ≲ δ−(p−1)η[Θ(rm)]p−1. For k and d > 0, we set v̄k = [ ln (Θ(rk)/2 + b vk + b ) ∨ 0 ] ∧ d. Applying Corollary 5.2 with a = Θ(rk)/2, b ∈ (0, 1) and d > 0, we have − ∫ B∗ k+1 |v̄k(x)− (v̄k)B∗ k+1 |p dx ≲ Rb,rk+1,rk((vk)−; 0) (6.7) where Rb,r,R(u−;x0) = 1 bp−1 ( r R )ps [TR(u−;x0)] p−1 + ( 1 + ∥V+∥Lq(Ω) ) . If we set b = δ ps p−1−ηΘ(rk) in (6.7), then by (5.19) and (6.6) we obtain that − ∫ B∗ k+1 |v̄k(x)− (v̄k)B∗ k+1 | dx ≤ ( − ∫ B∗ k+1 |v̄k(x)− (v̄k)B∗ k+1 |p dx )1/p ≤ c ( 1 + ∥V+∥Lq(Ω) )1/p (6.8) where c > 0 is a constant depending only on n, s, p, η, λ and Λ. From (6.4), we can derive the estimate d = 1 |B∗ k+1 ∩ {vk ≥ Θ(rk)/2}| ∫ B∗ k+1∩{vk≥Θ(rk)/2} d dx = 1 |B∗ k+1 ∩ {vk ≥ Θ(rk)/2}| ∫ B∗ k+1∩{v̄k=0} d dx ≤ 2 |B∗ k+1| ∫ B∗ k+1 (d− v̄k) dx = 2 ( d− (v̄k)B∗ k+1 ) . (6.9) The estimates (6.8) and (6.9) make it possible to obtain the estimate |B∗ k+1 ∩ {v̄k = d}| |B∗ k+1| d ≤ 2 |B∗ k+1| ∫ B∗ k+1∩{v̄k=d} ( d− (v̄k)B∗ k+1 ) dx ≤ 2 |B∗ k+1| ∫ B∗ k+1∩{v̄k=d} ( v̄k − (v̄k)B∗ k+1 ) dx ≲ ( 1 + ∥V+∥Lq(Ω) )1/p . (6.10) We now set d = d∗ := ln (Θ(rk)/2 + δ ps p−1−ηΘ(rk) 3 δ ps p−1−ηΘ(rk) ) . Then we see that d∗ ∼ ln(1/δ). By (6.10), we have |B∗ k+1 ∩ {vk ≤ 2 δ ps p−1−ηΘ(rk)}| |B∗ k+1| ≤ c ( 1 + ∥V+∥Lq(Ω) )1/p d∗ ≤ c0 ( 1 + ∥V+∥Lq(Ω) )1/p ln(1/δ) . (6.11) Now we proceed the next step with a well-known iteration process as follows. For i ∈ N ∪ {0}, we set ρi = (1 + 2−i)rk+1, ρ̄i = ρi + ρi+1 2 , Bi = Bρi , B̄i = Bρ̄i . For i ∈ N ∪ {0}, we consider a function ζi ∈ C∞ c (Bρ̄i) with ζi|Bρi+1 ≡ 1 such that 0 ≤ ζi ≤ 1 and |∇ζi| ≤ cρ−1 i in Rn. Moreover we set di = (1 + 2−i)δ ps p−1−ηΘ(rk) and wi = (di − vk)+ and Ni = |Bi ∩ {vk ≤ di}| |Bi| = |Bi ∩ {wi ≥ 0}| |Bi| . EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 23 From (6.11), we see that N0 ≤ c0 ( 1 + ∥V+∥Lq(Ω) )1/p ln(1/δ) . (6.12) By Theorem 1.5, we have [wiζi] p W s,p(Bρi ) ≲ ∫∫ Bρi ×Bρi [wi(x) ∨ wi(y)] p|ζi(x)− ζi(y)|p dK(x, y) + ( sup x∈supp(ζi) ∫ Rn\Bρi [wi(y)] p−1 K(x− y) dy ) ∥wiζ p i ∥L1(Bi) := A(ρi, wi, ζi) +B(ρi, wi, ζi). Then we have the estimate A(ρi, wi, ζi) ≲ dpi ∫ Bρi ∫ Bρi ∩{vk≤di} supRn |∇ζi|p |x− y|n+ps−p dx dy ≲ dpi ( 1 ρi )p ∫ Bρi ∩{vk≤di} ∫ B2ρi 1 |y|n+ps−p dy dx ≲ dpi ρ −ps i |Bi ∩ {vk ≤ di}|. (6.13) By the fact that |y − x| ≥ |y| − |x| ≥ ( 1− ρ̄i ρi ) |y| ≥ 2−i−2|y| for all y ∈ Rn\Bρi and x ∈ Bρ̄i , we obtain that B(ρi, wi, ζi) ≲ di2 i(n+ps)|Bi ∩ {vk ≤ di}| ∫ Rn\Bρi |wi(y)|p−1 |y|n+ps dy ≲ 2i(n+ps)diρ −ps i |Bi ∩ {vk ≤ di}| [Trk+1 (wi; 0)] p−1. (6.14) Thus it follows from (6.13) and (6.14) that [wiζi] p W s,p(Bρi ) ≲ ( dpi + 2i(n+ps) di[Trk+1 (wi; 0)] p−1 ) ρ−ps i |Bi ∩ {vk ≤ di}|. (6.15) From (6.6) and that wi ≤ 2δ ps p−1−ηΘ(rk) in Bk and wi ≤ |vk|+2δ ps p−1−ηΘ(rk) in Rn, we can derive that [Trk+1 (wi; 0)] p−1 ≲ rpsk+1 ∫ Bk\Bk+1 |wi(y)|p−1 |y|n+ps dy + (rk+1 rk )ps [Trk(wi; 0)] p−1 ≲ δps−(p−1)η [Θ(rk)] p−1 + δps[Trk(vk; 0)]p−1 ≲ dp−1 i . (6.16) Thus by (6.15) and (6.16), we have [wiζi] p W s,p(Bρi ) ≲ 2i(n+ps) dpi ρ −ps i |Bi ∩ {vk ≤ di}|. (6.17) By applying (6.17) and the fractional Sobolev’s inequality with exponent γ = n n− ps , we can deduce the inequalities(∫ Bi+1 |wi|pγ dx )1/γ ≤ (∫ Bi |wiζi| pn n−ps dx )n−ps n ≲ [wiζi] p W s,p(Bρi ) + ρ−ps i ∥wiζi∥pLp(Bρi ) ≲ 2i(n+ps) dpi ρ −ps i |Bi ∩ {vk ≤ di}|. (6.18) Since |Bi+1| ∼ ρni ∼ |Bi| and wi = (di − vk)+ ≥ (di − di+1)1{vk≤di+1} ≥ 2−i−2di 1{vk≤di+1}, 24 Y.-C. KIM EJDE-2025/83 estimate (6.18) yields that (di−di+1) p ( |Bi+1 ∩ {vk ≤ di+1| |Bi+1| )1/γ ≤ ρpsi |Bi| (∫ Bi+1 |wi|pγ dx dt )1/γ ≲ 2i(n+ps) dpi |Bi ∩ {vk ≤ di}| |Bi| , which gives N 1/γ i+1 ≤ c 2i(n+ps) dpi (di − di+1)p Ni ≤ c2i(n+ps+p)Ni. This leads us to Ni+1 ≤ c12 iγ(n+ps+p)N 1+ ps n−ps i . If we could show that N0 = |B0 ∩ {vk ≤ 2δ ps p−1−ηΘ(rk)}| |B0| ≤ c −n−ps ps 1 2 − γ(n−ps)2(n+ps+p) p2s2 := c∗, (6.19) then by Lemma 2.3 we conclude that limi→∞ Ni = 0, i.e. inf Bk+1 vk > δ ps p−1−ηΘ(rk). To guarantee (6.19), by (6.12) we observe that c∗ ≥ c0 ( 1 + ∥V+∥Lq(Ω) )1/p ln(1/δ) ⇔ δ ≤ e−(c0/c∗)(1+∥V+∥Lq(Ω)) 1/p , and so we choose δ > 0 as δ = e−(c0/c∗)(1+∥V+∥Lq(Ω)) 1/p ∧ (1 4 ) p−1 ps . If vk = v − infBk v, then by (6.1) we have oscBk+1 v = oscBk+1 vk ≤ oscBk v − inf Bk+1 vk ≤ (1− δ ps p−1−η)Θ(rk). (6.20) If vk = Θ(rk)− (v − infBk v), then we have oscBk+1 v = oscBk+1 vk = Θ(rk)− inf Bk+1 v + inf Bk v − inf Bk+1 vk ≤ (1− δ ps p−1−η)Θ(rk). (6.21) From (6.20) and (6.21), we obtain that oscBk+1 v ≤ (1−δ ps p−1−η)Θ(rk) = (1−δ ps p−1−η) ( rk rk+1 )η Θ(rk+1) = (1−δ ps p−1−η)δ−ηΘ(rk+1). (6.22) To find η so that (1− δ ps p−1−η)δ−η ≤ 1, we consider the function ξ(η) = δη + δ ps p−1−η. We note that ξ′(η) = δη ln δ ( 1− δ ps p−1−2η ) = 0 ⇔ η = ps 2(p− 1) := ηp,s , ξ′(η) < 0 for η < ηp,s and ξ′(η) > 0 for η > ηp,s. These facts imply that the graph of ξ is going down from the point (0, 1 + δ2ηp,s) to the point (ηp,s, 1 + 2δηp,s) and is going up the point (2ηp,s, 1 + δ2ηp,s) right after that. Since we see that 2δηp,s < 1 + δ2ηp,s and 2δηp,s < 1, we can find exactly two η’s inside (0, ps p−1 ) so that δη + δ ps p−1−η = 1. (6.23) If we set Y = δη, then the above equation (6.23) will be transformed into Y 2 − Y + δ ps p−1 Y = 0 and its solutions are δη = Y = 1± √ 1− 4δ ps p−1 2 ∈ (0, 1), EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 25 because δ < (1/4) p−1 ps . Hence it turns out that the solutions of (6.23) are η±0 = ln ( 1± √ 1−4δ ps p−1 2 ) ln δ . Then we see that ξ(η) ≥ 1, i.e. (1 − δ ps p−1−η)δ−η ≤ 1 for all η ∈ (0, η−0 ] ∪ [η+0 , 2ηp,s). Thus, if η ∈ (0, η−0 ] ∪ [η+0 , 2ηp,s), then by (6.22) we conclude that oscBk+1 v ≤ Θ(rk+1). Therefore we complete the proof. □ Acknowledgements. Yong-Cheol Kim was supported by School of Education, Korea University Grant in 2024. References [1] M. 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Introduction Notation 1.1. Remarks 2. Preliminaries 3. Weighted norm inequalities on the Ap-Muckenhoupt class 4. Local properties of weak subsolutions 5. Logarithm of a weak solution is a locally bounded mean oscillation function 6. Interior Hölder regularity Acknowledgements References