Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 69, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.69 LOWER BOUNDS AT INFINITY FOR SOLUTIONS TO SECOND ORDER ELLIPTIC EQUATIONS TU NGUYEN Abstract. We study lower bounds at infinity for solutions to |Pu| ≤M |x|−δ1 |∇u|+ M |x|−δ0 |u| where P is a second order elliptic operator. Our results are of quantitative nature and generalize those obtained in [3, 6]. 1. Introduction Let Pu = div(A∇u) be a second order elliptic operator in divergence form, where A is symmetric, Lipschitz and uniformly elliptic, i.e. λ|ξ|2 ≤ 〈A(x)ξ, ξ〉 ≤ λ−1|ξ|2 ∀x, ξ ∈ Rn, for some λ ∈ (0, 1]. In this paper we study lower bounds at infinity for (non-trivial) solutions of |Pu| ≤M |x|−δ1 |∇u|+M |x|−δ0 |u| in Rn where M > 0 and δ0, δ1 ∈ R. The first result in this direction was obtained by Meshkov [7] for solutions of |∆u| ≤M |u| in Rn. (1.1) He showed that if u(x) exp(C|x|4/3) ∈ L∞(Rn) for all C > 0 then u must vanish identically. The exponent 4/3 is optimal as Meshkov also constructed a function u : R2 → C satisfying (1.1) such that |u(x)| ≤ C exp(−|x|4/3) ∀x ∈ R2. This result was later extended in [2] to solutions of |∆u+ Eu| ≤M |x|−δ0 |u| ∀x ∈ Rn (1.2) where E is a real constant. It was proved that if u(x) exp(C|x|α) ∈ L∞(Rn) for all C > 0 where α = max{1, 4−2δ03 } then u ≡ 0. The optimality of the exponent α was shown by a variant of Meshkov’s example. 2020 Mathematics Subject Classification. 35J15, 35A02. Key words and phrases. Carleman estimates; strong unique continuation. ©2023. This work is licensed under a CC BY 4.0 license. Submitted September 4, 2023. Published October 16, 2023. 1 2 T. NGUYEN EJDE-2023/69 The first quantitative version of these results was obtained by Bourgain and Kenig [1]. They proved that if u satisfies (1.1) then there exists C > 0 such that∫ B(x,1) u2 ≥ exp(−C|x|4/3 log |x|) ∀|x| ≥ 10. Their method can be easily adapted to give the bound∫ B(x,|x|/2) u2 ≥ exp(−C|x|4/3) ∀|x| ≥ 10, from which Meshkov’s qualitative result can be recovered. Subsequently, [3, 6] extended these estimates to solutions of operators with lower order terms, and operators with variable second order coefficients. The main results of this paper are improvements of their results. Theorem 1.1. Let P be as above and u be a nontrivial solution of |Pu| ≤M |x|−δ1 |∇u|+M |x|−δ0 |u| in Rn \B1 (1.3) where M > 0 and δ1, δ2 ∈ R. Suppose that α ≥ max{ 4−2δ03 , 2−2δ1} and α > 1−λ2. Then there exists ε = ε(n, α, λ) > 0 such that if |∇A(x)| ≤ ε |x| and |u(x)| ≤ eM |x| α when |x| > 1, then there exists C > 0 so that if |x| ≥ 10,∫ B(x,|x|/2) u2 ≥ exp(−C|x|α) (1.4) and ∫ B(x,1) u2 ≥ exp(−C|x|α log |x|). (1.5) We also have the following similar result for operators with differentiable poten- tials. Theorem 1.2. Let P be as above and V be a Lipschitz function satisfying |V |+ |Ax · ∇V | ≤M |x|−δ2 . Let u be a nontrivial solution of |Pu+ V u| ≤M |x|−δ1 |∇u|+M |x|−δ0 |u| in Rn \B1. Suppose that α ≥ max{ 4−2δ03 , 2 − 2δ1, 2−δ2 2 } and α > 1 − λ2. Then there exists ε = ε(n, α, λ) > 0 such that if |∇A(x)| ≤ ε |x| and |u(x)| ≤ eM |x| α when |x| > 1 then there exists C > 0 so that if |x| ≥ 10,∫ B(x,|x|/2) u2 ≥ exp(−C|x|α) and ∫ B(x,1) u2 ≥ exp(−C|x|α log |x|). When V is a constant, we can take δ2 = 0 to obtain the following generalization of (1.2). EJDE-2023/69 LOWER BOUNDS FOR SECOND ORDER ELLIPTIC EQUATIONS 3 Corollary 1.3. Let E be a real number and u is a nontrivial solution of |Pu+ Eu| ≤M |x|−δ1 |∇u|+M |x|−δ0 |u| in Rn \B1. Let α = max{1, 4−2δ03 , 2− 2δ1}, then there exists ε = ε(n, α, λ) > 0 such that if |∇A(x)| ≤ ε |x| and |u(x)| ≤ eM |x| α when |x| > 1 then there exists C > 0 so that if |x| ≥ 10,∫ B(x,|x|/2) u2 ≥ exp(−C|x|α) and ∫ B(x,1) u2 ≥ exp(−C|x|α log |x|). In [6], similar estimates were obtained under the slightly stronger condition |∇A(x)| ≤ C |x|1+ε for |x| > 1. This was proved by using a chain of balls argument with balls of different radii which lead to slightly weaker lower bounds of the form exp(−C|x|α(log |x|)γ(x)), where γ(x) is a function of log |x|. Besides giving stronger, and perhaps optimal, results, our approach also works in the case α < 1, which is not possible with the method of [6]. We are not sure if the condition α > 1− λ2 is necessary though. Our proof, detailed in Section 3, starts by proving the following lower bound for u on annuli ∫ {R−1≤|x|≤R} u2 ≥ exp(−CRα) ∀R ≥ 10. Using this bound and a chain of balls argument with balls of the same size we obtain (1.4). The key ingredient in this step is a three-ball inequality (see (3.6) and (3.7)). Finally, we deduce (1.5) from (1.4) using another application of the three-ball inequality. 2. Carleman estimates In this section, we collect some Carleman estimates that play the key role in the proofs of Theorems 1.1 and 1.2. Throughout this section, C denotes a constant that only depends on α, n, λ and Λ, whose value may change from line to line. The first Carleman estimate is a generalization of an estimate in [7, Lemma 1] where it is proved for P = ∆ and α ∈ (2/3, 2]. For convenience, we will use the notations r = |x| and ‖ξ‖2 = 〈A(x)ξ, ξ〉. Proposition 2.1. Let P be as in Theorem 1.1, and α > 1− λ2. Then there exist positive constants β0, C0 and ε depending on α, n, λ and Λ such that if |∇A(x)| ≤ ε/|x| for |x| ≥ 1, then β3 ∫ r2α−2e2βr α u2 + β ∫ e2βr α |∇u|2 ≤ C0 ∫ r2−αe2βr α |Pu|2, (2.1) 4 T. NGUYEN EJDE-2023/69 for u ∈ C∞c (Rn \B1) and β ≥ β0. Proof. Let v = eβr α u then eβr α Pu = eβr α P (e−βr α v) =: Pβv. Since eβr α |∇u| ≤ |∇v| + αβrα−1|v|, it follows that (2.1) is equivalent to (with possibly a diffenrent value of C0) β3 ∫ r2α−2v2 + β ∫ |∇v|2 ≤ C0 ∫ r2−α|Pβv|2. (2.2) We have Pβv = [div(A∇v) + α2β2r2α−4‖x‖2v] − αβ [ 2rα−2〈Ax,∇v〉+ div(rα−2Ax)v ] . Let F = trA+ (α− 2) ‖x‖2 |x|2 + |Ax|2 ‖x‖2 − λ and P̃βv = [div(A∇v) + α2β2r2α−4‖x‖2v] − αβ[2rα−2〈Ax,∇v〉+ rα−2Fv] =: Mv −Nv. Since |div(rα−2Ax)− rα−2F | ≤ Cαβrα−2, if β ≥ 4C2α2, then∫ |Pβv − P̃βv|2r2−α ≤ C2α2β2 ∫ rα−2v2 ≤ 1 4 β3 ∫ r2α−2v2. Thus, it suffices to prove (2.2) with Pβv replaced by P̃βv. We have∫ |P̃βv|2r2−α ≥ −2 ∫ r2−αMvNv = −2α3β3 ∫ r2α−4‖x‖2[2〈Ax,∇v〉v + Fv2] − 2αβ ∫ 2 div(A∇v)〈Ax,∇v〉+ F div(A∇v)v =: I + II. (2.3) Writing v∇v = 1 2∇(v2) then integrating by parts, we obtain I = 2α3β3 ∫ [div(r2α−4‖x‖2Ax)− r2α−4F‖x‖2]v2 = 2α3β3 ∫ [2(α− 2) ‖x‖2 |x|2 + 2 |Ax|2 ‖x‖2 − F + trA]‖x‖2r2α−4v2 + 2α3β3 ∫ [∂iaijxj‖x‖2r2α−4 + aikxk∂iajlxlxjr 2α−4]v2 ≥ 2α3β3 ∫ [(α− 2) ‖x‖2 |x|2 + |Ax|2 ‖x‖2 + λ− Cε]‖x‖2r2α−4v2. (2.4) To estimate the term II in (2.3), we first use the Rellich-Necas identity to write 2 div(A∇v)〈Ax,∇v〉 = 2alkxk∂i(aij∂jv∂lv)− alkxkaij∂l(∂iv∂jv). Then integrating by parts, we obtain II = 2αβ ∫ 2∂i(alkxk)aij∂jv∂lv − ∂l(alkxkaij)∂iv∂jv +A∇v · ∇(Fv) EJDE-2023/69 LOWER BOUNDS FOR SECOND ORDER ELLIPTIC EQUATIONS 5 = 2αβ ∫ 2|A∇v|2 + (F − trA)‖∇v‖2 + vA∇v · ∇F + 2αβ ∫ 2∂ialkxkaij∂jv∂lv − ∂l(alkaij)xk∂iv∂jv. As |∇F | ≤ C/|x|, we have |vA∇v · ∇F | ≤ Cε‖∇v‖2 + ε−1r−2v2. Together with |A∇v|2 ≥ λ‖∇v‖2, this implies II ≥ 2αβ ∫ ( (α− 2) ‖x‖2 |x|2 + |Ax|2 ‖x‖2 + λ− Cε ) ‖∇v‖2 − 2αβε−1 ∫ r−2v2. (2.5) We have (α− 2) ‖x‖2 |x|2 + |Ax|2 ‖x‖2 + λ ≥ (α− 1) ‖x‖2 |x|2 + λ ≥ min{αλ, (α− 1)λ−1 + λ} =: µ > 0. Thus, if ε = µ 2C and β ≥ 4C αµ , then from (2.4) and (2.5) we obtain∫ |P̃βv|2r2−α ≥ µα3β3 ∫ r2α−2v2 + µαβ ∫ ‖∇v‖2 − 2αβε−1 ∫ r−2v2 ≥ 1 2 µα3β3 ∫ r2α−2v2 + µαβ ∫ ‖∇v‖2. This completes the proof. � Proposition 2.2. Let P be as in Theorem 1.2, V be a Lipschitz function and α > 1 − λ2. Then there exist positive constants β0, C0 and ε depending on α, n, λ and Λ such that if A satisfies |∇A(x)| ≤ ε/|x| for |x| ≥ 1, then β3 ∫ r2α−2e2βr α u2 + β ∫ e2βr α |∇u|2 ≤ C0 ∫ r2−αe2βr α |Pu|2 + C0β ∫ e2βr α (|V |+ |Ax · ∇V |)u2. for u ∈ C∞c (Rn \B1) and β ≥ β0. Proof. The proof is very similar to that of the previous proposition, so we will only indicate the modifications needed. Now P̃βv has one more term V v which we incorporate into Mv, i.e. P̃βv = [div(A∇v) + α2β2r2α−4‖x‖2v + V v] − αβ[2rα−2〈Ax,∇v〉+ rα−2Fv] =: Mv −Nv. We then have −2 ∫ r2−αMvNv = I + II + III where I and II are as in (2.3) and III = −4αβ ∫ 〈Ax,∇v〉V v − 2αβ ∫ FV v2 = 2αβ ∫ [div(V Ax)− FV ]v2 6 T. NGUYEN EJDE-2023/69 As |div(V Ax)− FV | ≤ C(|V |+ |Ax · ∇V |), |III| ≤ Cβ ∫ (|V |+ |Ax · ∇V |)v2 and the proposition follows. � The next Carleman estimate is rather standard and can be found in [4, 5]. Proposition 2.3. Let Pu = div(A∇u) where A is elliptic, symmetric with A(0) = I and |∇A| ≤ 1/ √ λ. There exist positive constants β0, C0, and ρ ≤ √ λ/4 depending only on n and an increasing function w satisfying 1 C0 ≤ w(x) |x| ≤ C0 such that β3 ∫ w−1−2βu2 + β ∫ w1−2β |∇u|2 ≤ C0 ∫ w2−2β |Pu|2 (2.6) for u ∈ C2 c (Bρ \ {0}) and β ≥ β0. We also have a similar estimate for operators with differentiable potentials. Proposition 2.4. Let Pu = div(A∇u) + V u where V is Lipschitz, A is elliptic, symmetric with A(0) = I and |∇A| ≤ 1/ √ λ. There exist positive constants β0, C0 and ρ ≤ √ λ/4 depending only on n and an increasing function w satisfying 1 C0 ≤ w(x) |x| ≤ C0 such that β3 ∫ w−1−2βu2 + β ∫ w1−2β |∇u|2 ≤ C0 ∫ w2−2β |Pu|2 + C0β ∫ w1−2β(|V |+ |Ax · ∇V |)u2 for u ∈ C2 c (Bρ \ {0}) and β ≥ β0. 3. Proof of main theorem The proofs of Theorems 1.1 and 1.2 follow the same lines, using Propositions 2.2 and 2.4 for the first theorem and Propositions 2.1 and 2.3 for the second theorem. Throughout this section, C denotes a constant that only depends on α, δ1, δ2, λ,M and n, whose value may change from line to line. For r2 > r1 > 0, we let Ar1,r2 = {x ∈ Rn : r1 ≤ |x| ≤ r2}. The ball of radius r centered at a is denoted by B(a, r) and Br = B(0, r). As indicated in the Introduction, we first prove a lower bound for the L2-norm of u on annuli. Lemma 3.1. Let P , u and α be as in the statement of Theorem 1.1. Then there exists positive constant C1 such that∫ AR−1,R u2 ≥ exp(−C1R α) ∀R ≥ 10. (3.1) Proof. Let ϕ be a smooth cut-off function satisfying ϕ(x) = { 1 if 5 3 ≤ |x| ≤ R− 2 3 0 if 4 3 ≥ |x| or |x| ≥ R− 1 3 and |∇ϕ(x)|+ |∇2ϕ(x)| ≤ Cn ∀x. EJDE-2023/69 LOWER BOUNDS FOR SECOND ORDER ELLIPTIC EQUATIONS 7 Let v = ϕu and E = sppt(∇ϕ) ⊂ A4/3,5/3 ∪ AR−2/3,R−1/3. We have |Pv| = |ϕPu+ 2A∇u · ∇ϕ+ div(A∇ϕ)u| ≤Mϕ ( |x|−δ0 |u|+ |x|−δ1 |∇u| ) + C[|∇u|+ |u|]1E ≤M ( |x|−δ0 |v|+ |x|−δ1 |∇v| ) + C[|∇u|+ ( 1 +M |x|−δ1 ) |u|]1E . Applying the Carleman estimate (2.1), we obtain for β ≥ β0, β3 ∫ e2βr α r2α−2v2 + β ∫ e2βr α |∇v|2 ≤ C0 ∫ e2βr α r2−α|Pv|2 ≤ 4C0M 2 ∫ e2βr α ( r2−α−2δ0v2 + r2−α−2δ1 |∇v|2 ) + 4C0C 2(M2 + 1) ∫ E e2βr α [|∇u|2 + ( 1 + r−2δ1 ) u2]. Choose β = β0 + 8C0M 2, then the first term of the right-hand side is absorbed by the left-hand side since r2−α−2δ0 ≤ r2α−2 and r2−α−2δ1 ≤ 1 for r ≥ 1. As v = u on A3,7, we deduce that e2β·3 α ∫ A3,7 u2 ≤ β3 ∫ e2βr α r2α−2v2 ≤ C ∫ E e2βr α [|∇u|2 + ( 1 + r−2δ1 ) u2]. (3.2) Applying the standard Cacciopolli’s inequality, we obtain∫ E e2βr α [|∇u|2 + ( 1 + r−2δ1 ) u2] ≤ Ce2β·2 α ∫ A1,2 u2 + CRδe2βR α ∫ AR−1,R u2, where δ = max{−δ0,−2δ1, 0}. Combining the above inequality and (3.2), we obtain e2β·3 α ∫ A3,7 u2 ≤ Ce2β·2 α ∫ A1,2 u2 + CRδe2βR α ∫ AR−1,R u2. (3.3) If β ≥ log(2C ∫ A1,2 u2/ ∫ A3,7 u2) 2(3α − 2α) , then the first term of the right-hand side of (3.3) can be absorbed by the left-hand side, and we obtain R−δe−2βR α ∫ A3,7 u2 ≤ C ∫ AR−1,R u2dx. This completes the proof. � Next, we show that (3.1) and the upper bound on u give the desired lower bounds. Lemma 3.2. Let P , u be as in the statement of Theorem 1.1 and τ = √ λρ 4 where ρ is the constant appears in the statement of Proposition 2.3. Assume that for some positive constants C1 and M ,∫ AR−1,R u2 ≥ e−C1R α ∀R ≥ 10, 8 T. NGUYEN EJDE-2023/69 and |u(x)| ≤ eM |x| α ∀|x| ≥ 1. Then there exists C2 > 0 such that if |x| = R ≥ 10, then∫ B(x,τR) u2 ≥ e−C2R α (3.4) and ∫ B(x,1) u2 ≥ e−C2R α logR. (3.5) Proof. We first prove a version of the standard three-ball inequality: there exists C > 0 such that for a ∈ AR−τR,R+τR and 1/R ≤ τ0 ≤ τ1/λ ≤ τ2/2 ≤ 2τ/λ, we have either ∫ B(a,τ1R) u2 ≤ (w( √ λτ2/2) w( √ λτ0/2) )CRα ∫ B(a,τ0R) u2 (3.6) or ∫ B(a,τ1R) u2 ≤ CRδ (∫ B(a,τ0R) u2 )θ(∫ B(a,τ2R) u2 )1−θ . (3.7) Here δ = max{1, 2− δ0, 2− 2δ1} and θ = log(w( √ λτ2/2))− log(w(τ1/ √ λ)) log(w( √ λτ2/2))− log(w( √ λτ0/2)) . To show this, we first make a change of variables. Let S = A(a)−1/2, AR(x) = SA(a + S−1Rx)St and PRv = div(AR∇v). Then for uR(x) = u(a + S−1Rx), we have PRuR(x) = R2Pu(a+ S−1Rx). Thus, |PRuR| ≤ CR1−δ1 |∇uR|+ CR2−δ0 |uR| in Bρ. Let r0 = √ λτ0, r1 = τ1/ √ λ, r2 = √ λτ2 and ϕ be a smooth cut-off function satisfying ϕ(x) = { 1 if 2 3r0 ≤ |x| ≤ 1 2r2 0 if 1 2r0 ≥ |x| or |x| ≥ 2 3r2 and |∇lϕ(x)| ≤ C|x|−l, l = 1, 2. Let v = ϕuR and E = sppt(∇ϕ) ⊂ Ar0/2,2r0/3 ∪ Ar2/2,2r2/3. We have |PRv| = |ϕPRuR + 2AR∇uR · ∇ϕ+ div(AR∇ϕ)uR| ≤ Cϕ ( R2−δ0 |uR|+R1−δ1 |∇uR| ) + C[|x|−1|∇uR|+ |x|−2|uR|]1E ≤ C ( R2−δ0 |v|+R1−δ1 |∇v| ) + C[|x|−1|∇uR|+ ( |x|−2 + |x|−1R1−δ1 ) |uR|]1E . Note that AR(0) = Id and |∇AR| ≤ √ λ in Bρ. Thus, we can apply Proposition 2.3 to PR and v to obtain for β ≥ β0 β3 ∫ w−1−2βv2 + β ∫ w1−2β |∇v|2 EJDE-2023/69 LOWER BOUNDS FOR SECOND ORDER ELLIPTIC EQUATIONS 9 ≤ C ∫ w2−2β |PRv|2 ≤ C ∫ w2−2β (R4−2δ0v2 +R2−2δ1 |∇v|2 ) + C ∫ E w2−2β [|x|−2|∇uR|2 + ( |x|−4 + |x|−2R2−2δ1 ) u2R]. If β ≥ CRα then the first term of the right-hand side is absorbed by the left-hand side, hence we obtain β3 ∫ w−1−2βv2 ≤ C ∫ E w2−2β [|x|−2|∇uR|2 + ( |x|−4 + |x|−2R2−2δ1 ) u2R]. Since v = u on Ar0,r1 the left-hand side is greater than w−1−2β(r1) ∫ Ar0,r1 u2R. By the standard Cacciopolli’s inequality, the right-hand side is smaller than CRδw−1−2β(r0/2) ∫ A r0 3 ,r0 u2R + CRδw−1−2β(r2/2) ∫ A r2 3 ,r2 u2R, where δ = max{1, 2− δ0, 2− 2δ1}. Thus, we obtain w−1−2β(r1) ∫ Ar0,r1 u2R ≤ CRδw−1−2β(r0/2) ∫ A r0 3 ,r0 u2R + CRδw−1−2β(r2/2) ∫ A r2 3 ,r2 u2R. Adding w−1−2β(r1) ∫ Br0 u2R to both sides gives w−1−2β(r1) ∫ Br1 u2R ≤CRδw−1−2β(r0/2) ∫ Br0 u2R (3.8) + CRδw−1−2β(r2/2) ∫ Br2 u2R. (3.9) If log( ∫ Br2 u2R/ ∫ Br0 u2R) 2 log(w(r2/2)/w(r0/2)) ≤ CRα then (3.6) holds. Otherwise, choosing β = log( ∫ Br2 u2R/ ∫ Br0 u2R) 2 log(w(r2/2)/w(r0/2)) in (3.8) gives w−1−2β(r1) ∫ Br1 u2R ≤ CRδw−1−2β(r0/2) ∫ Br0 u2R, which implies ∫ Br1 u2R ≤ CRδ (∫ Br0 u2R )θ(∫ Br2 u2R )1−θ . Undoing the change of variables, noting that a+S−1RBrj ⊂ B(a, τjR) for j = 0, 2 while a+ S−1RBr1 ⊃ B(a, τ1R), we obtain (3.7). 10 T. NGUYEN EJDE-2023/69 Choose τ0 = τ , τ1 = 2τ , and τ2 = 4τ/λ. It is easy to see that with this choice of parameters, both (3.6) and (3.7) implies that for some C3 > 0,∫ B(a,2τR) u2 ≤ eC3R α (∫ B(a,τR) u2 )θ if a ∈ AR−τR,R+τR. (3.10) Here we have used the upper bound |u(x)| ≤ eM |x|α . We next deduce (3.4) from (3.10). Let |x| = R and Q = {a1 = x, a2, a3, . . . , aN} be a τR 2 -net on the sphere of radius R. By scaling and symmetry, it easy to check that this can be done with N independent of R and x. Note that any aj ∈ Q can be connected to a1 = x by a sequence of points in Q such that the distances between consecutive points smaller than τR. Thus, applying (3.10) repeatedly gives∫ B(aj ,τR) u2 ≤ eC3R α/(1−θ) (∫ B(x,τR) u2 )θN . Since ∪Nj=1B(aj , τR) ⊃ AR−1,R, summing over j gives e−C1R α ≤ ∫ AR−1,R u2 ≤ N∑ j=1 ∫ B(aj ,τR) u2 ≤ NeC3R α/(1−θ) (∫ B(x,τR) u2 )θN from which (3.4) follows. Finally, we prove the lower bound (3.5). Choose τ0 = 1/R, τ1 = 2τ and τ2 = 4τ/λ. If (3.6) holds then (3.5) follows because w( √ λτ2/2) w( √ λτ0/2) = w(ρ/2) w( √ λ/(2R)) ≤ CR. On the other hand, if (3.7) holds, then it follows that∫ B(x,τR) u2 ≤ CRδeMRα (∫ B(x,1) u2 )θ . Since θ = log(w(ρ/2))− log(w(ρ/4)) log(w(ρ/2))− log(w( √ λ/(2R))) ≥ C logR , inequality (3.5) follows from (3.4). � Acknowledgments. The author wants to thank the referee for several helpful comments. This work is supported by Vietnam National University, Ho Chi Minh City, under grant number C2021-28-03. References [1] J. Bourgain, C. E. Kenig; On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math., 161(2):389–426, 2005. [2] J. Cruz-Sampedro; Unique continuation at infinity of solutions to schrödinger equations with complex-valued potentials. Proc. Edinburgh Math. Soc. (2), 42(1):143–153, 1999. [3] B. Davey; Some quantitative unique continuation results for eigenfunctions of the magnetic schrödinger operator. Comm. Partial Differential Equations, 39(5):876–945, 2014. [4] L. Escauriaza, S. Vessella; Optimal three cylinder inequalities for solutions to parabolic equa- tions with lipschitz leading coefficients. Comtemp. Math., 333:79–87, 2003. [5] L. Hömander; Uniqueness theorems for second order elliptic differential equations. Comm. P.D.E., 8:21–64, 1983. [6] C.-L. Lin, J.-N. Wang; Quantitative uniqueness estimates for the general second order elliptic equations. J. Funct. Anal., 266(8):5108–5125, 2014. EJDE-2023/69 LOWER BOUNDS FOR SECOND ORDER ELLIPTIC EQUATIONS 11 [7] V. Z. Meshkov; On the possible rate of decay at infinity of solutions of second order partial differential equations. Matematicheskii Sbornik, 182(3):364–383, 1991. Tu Nguyen Department of Mathematics, International University, Vietnam National University, Ho Chi Minh City, Vietnam Email address: natu@hcmiu.edu.vn 1. Introduction 2. Carleman estimates 3. Proof of main theorem Acknowledgments References