LOWER BOUNDS AT INFINITY FOR SOLUTIONS TO SECOND ORDER ELLIPTIC EQUATIONS

. We study lower bounds at inﬁnity for solutions to


Introduction
Let P u = div(A∇u) be a second order elliptic operator in divergence form, where A is symmetric, Lipschitz and uniformly elliptic, i.e.where M > 0 and δ 0 , δ 1 ∈ R.
The first result in this direction was obtained by Meshkov [7] for solutions of He showed that if u(x) exp(C|x| 4/3 ) ∈ L ∞ (R n ) for all C > 0 then u must vanish identically.The exponent 4/3 is optimal as Meshkov also constructed a function u : R 2 → C satisfying (1.1) such that This result was later extended in [2] to solutions of where E is a real constant.It was proved that if u(x) exp(C|x| α ) ∈ L ∞ (R n ) for all C > 0 where α = max{1, 4−2δ0 3 } then u ≡ 0. The optimality of the exponent α was shown by a variant of Meshkov's example.
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 where M > 0 and We also have the following similar result for operators with differentiable potentials.
Theorem 1.2.Let P be as above and V be a Lipschitz function satisfying Let u be a nontrivial solution of when |x| > 1 then there exists C > 0 so that if |x| ≥ 10, When V is a constant, we can take δ 2 = 0 to obtain the following generalization of (1.2).
Corollary 1.3.Let E be a real number and u is a nontrivial solution of In [6], similar estimates were obtained under the slightly stronger condition 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 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 − λ 2 is necessary though.Our proof, detailed in Section 3, starts by proving the following lower bound for u on annuli 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.

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.
Proposition 2.2.Let P be as in Theorem 1.2, V be a Lipschitz function and α > 1 − λ 2 .Then there exist positive constants β 0 , C 0 and ε depending on α, n, λ and Λ such that if A satisfies ) and β ≥ β 0 .Proof.The proof is very similar to that of the previous proposition, so we will only indicate the modifications needed.Now P β v has one more term V v which we incorporate into M v, i.e.
We then have −2 r 2−α M vN v = I + II + III where I and II are as in (2.3) and and the proposition follows.
The next Carleman estimate is rather standard and can be found in [4,5].

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, δ 1 , δ 2 , λ, M and n, whose value may change from line to line.For r 2 > r 1 > 0, we let A r1,r2 = {x ∈ R n : r 1 ≤ |x| ≤ r 2 }.The ball of radius r centered at a is denoted by B(a, r) and B r = B(0, r).
As indicated in the Introduction, we first prove a lower bound for the L 2 -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 C 1 such that Proof.Let ϕ be a smooth cut-off function satisfying Let v = ϕu and We have Applying the Carleman estimate (2.1), we obtain for β ≥ β 0 , Choose β = β 0 + 8C 0 M 2 , then the first term of the right-hand side is absorbed by the left-hand side since r 2−α−2δ0 ≤ r 2α−2 and r 2−α−2δ1 ≤ 1 for r ≥ 1.As v = u on A 3,7 , we deduce that Applying the standard Cacciopolli's inequality, we obtain where δ = max{−δ 0 , −2δ 1 , 0}.Combining the above inequality and (3.2), we obtain then the first term of the right-hand side of (3.3) can be absorbed by the left-hand side, and we obtain This completes the proof.
Next, we show that (3.1) and the upper bound on u give the desired lower bounds.Then there exists and Proof.We first prove a version of the standard three-ball inequality: there exists Here δ = max{1, 2 − δ 0 , 2 − 2δ 1 } and To show this, we first make a change of variables.Let S = A(a and ϕ be a smooth cut-off function satisfying We have Thus, we can apply Proposition 2.3 to P R and v to obtain for β ≥ β 0 If β ≥ CR α then the first term of the right-hand side is absorbed by the left-hand side, hence we obtain Since v = u on A r0,r1 the left-hand side is greater than By the standard Cacciopolli's inequality, the right-hand side is smaller than where δ = max{1, 2 − δ 0 , 2 − 2δ 1 }.Thus, we obtain Adding w −1−2β (r 1 ) Br 0 u 2 R to both sides gives Undoing the change of variables, noting that a + S −1 RB rj ⊂ B(a, τ j R) for j = 0, 2 while a + S −1 RB r1 ⊃ B(a, τ 1 R), we obtain (3.7).
Choose τ 0 = τ , τ 1 = 2τ , and τ 2 = 4τ /λ.It is easy to see that with this choice of parameters, both (3.6) and (3.7) implies that for some C 3 > 0, B(a,2τ R) (3.10) Here we have used the upper bound |u(x)| ≤ e M |x| α .We next deduce (3.4) from (3.10).Let |x| = R and Q = {a 1 = x, a 2 , a 3 , . . ., a N } be a τ R 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 a j ∈ Q can be connected to a 1 = x by a sequence of points in Q such that the distances between consecutive points smaller than τ R. Thus, applying (3.10) repeatedly gives

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 C 1 and M , A R−1,R u 2 ≥ e −C1R α ∀R ≥ 10, and |u(x)| ≤ e M |x| α ∀|x| ≥ 1.