Generalized Harnack inequality for semilinear elliptic equations

This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.


Introduction
In this paper we study nonnegative weak solutions of the equation (1.1) div(A(x)Du) = f (u).
The coefficient matrix A(x) is assumed to be measurable and to satisfy the uniform ellipticity condition λ|ξ| 2 ≤ A(x)ξ, ξ and |A(x)ξ| ≤ Λ|ξ| for every ξ ∈ R n , where 0 < λ ≤ Λ.The function f : R → [0, ∞) is assumed to be nonnegative and nondecreasing.Throughout the paper we denote the integral function of f by F , i.e., F (t) = ˆt 0 f (s) ds.
Our main result reads as follows.
Theorem 1.1.Let u ∈ W 1,2 (B 2 ) be a nonnegative solution of (1.1).Denote M = sup B 1 u and m = inf B 1 u.There is a constant C which depends only on the ellipticity constants λ, Λ and the dimension n such that In particular, C is independent of the solution u and the function f .
Theorem 1.1 is in the same spirit as [10] for nondivergence equations with nonhomogeneous gradient drift term.The main difference from the previous result is that here we do not need any regularity nor growth assumptions on f , and most importantly, the constant in Theorem 1.1 is independent of f .In particular, the estimate is stable under scaling.
Harnack inequality for linear elliptic equations by Moser [17] is one of the most important result in the theory of elliptic partial differential equations.There are numerous generalizations of this theorem from which the most relevant for this paper is by Di Benedetto-Trudinger [8] who proved the Harnack inequality for quasiminimizers of integral functionals.Harnack inequality for general quasilinear equations has been considered e.g. by Serrin [24] (see also [14] for the Hölder continuity).This result has the disadvantage that the constant will depend on the solution itself.In Theorem 1.1 the inequality (1.2) is not in the classical form but its structure depends on the scaling of the equation.In this way the constant becomes independent of the solution.Theorem 1.1 is more similar to Pucci-Serrin [21] who introduced Harnack inequality in R 2 for quasilinear equations similar to (1.1) where the operator is allowed to be nonlinear but not to have dependence on x.Compared to [21] the advantage of Theorem 1.1 is that the dependence on every parameter in (1.2) is explicit, and we do not need any further assumption on f other than monotonicity.This makes the result more general and the esimate (1.2) more stable.Moreover, the result holds in any dimension and the operator is allowed to have just bounded coefficients.In particular, it is not possible to prove Theorem 1.1 by comparison argument.
The drawback of (1.2) is that it is in implicit form and it may be difficult to find the explicit relation between the maximum and the minimum.On the other hand (1.2) is a natural generalization of the classical Harnack inequality for equations of type (1.1).We illustrate this by giving a complete quantitative solution to the following problems.Here u ∈ W 1,2 (B 2 ) is a nonnegative solution of (1.1) with M = sup B 1 u and m = inf B 1 u.
(i) Strong minimum principle.If u is zero at one point in B 1 , is it zero everywhere?(ii) Boundedness.Is u bounded in B 1 by a constant which depends only on u(0)?(iii) Local Boundedness.Is there a radius r > 0 such that u is bounded in B r by a constant which depends only on u(0)?
The problem (i) has been considered by Vazquez [26] and by Pucci-Serrin and their collaborators [19,20,22,23] and we know that the strong minimum principle holds if f is positive and In fact, this condition is also necessary [3,7].Theorem 1.1 is in accordance with this and it provides a quantification of the strong minimum principle.By this we mean that if the integral of ( F (t) + t) −1 over (0, 1) diverges, then there is a continuous, increasing function Φ : [0, 1] → [0, 1] such that for M ≤ 1 it holds m ≥ Φ(M ), (see also [21]).In order to quantify the strong minimum principle we need to replace the function F (t) in the denumerator by F (t) + t.This is natural since in the linear case F ≡ 0 the estimate (1.2) then reduces to the classical Harnack inequality.
Similarly we find an answer to (ii).If the integral of ( , (see also [21]).This condition is very similar to the Keller-Osserman condition [11,18] which states that if f is positive and u is a nonnegative and nontrivial solution of (1.1) in the whole R n then For the proof see [13] or Theorem 1.2 below.The above condition is sharp.If f does not satisfy (1.3) then the solutions are not uniformly bounded, i.e., there exists a sequence of nonnegative solutions (u k ) of (1.1) such that u k (0) ≤ 1 and sup B 1 u k → ∞ as k → ∞.We leave this to the reader.Finally to answer (iii) we find that if u and f are as in Theorem 1.1, then we may always find a radius r > 0 such that sup Br u is uniformly bounded by a constant which depends on the value u(0).Indeed, since the constant in Theorem 1.1 does not depend on u and f , we obtain by a simple scaling argument that for M r = sup Br u it holds Note that when r → 0 the above estimate converges to the classical Harnack inequality.Therefore when r > 0 is small enough we have that M r < ∞.This result can be seen as a weak counterpart of the short time existence result for the one dimensional initial value problem y ′′ = f (y), y(0) = y 0 and y ′ (0) = y ′ 0 .Under the condition (1.3) the above initial value problem has a solution in the whole R.
Note that since f is nonnegative we have that any solution of (1.1) is a weak subsolution of the corresponding linear equation.Therefore by De Giorgi theorem [6] nonnegative solutions are locally bounded and by the De Giorgi-Nash-Moser theorem they are Hölder continuous.However, both the L ∞ -bound and the Hölder norm depend on the L 2 -norm of the solution and it is not clear how one can bound the L 2 -norm by knowing the value of the solution only at one point.
The statement of Theorem 1.1 is sharp and this can be seen already in dimension one.We leave this to the reader.The assumption f ≥ 0 is also necessary, i.e., Theorem 1.1 is not true for equations −∆u = f (u) where f is nonnegative and nondecreasing.This can be seen by a simple example which we give at the end of the paper.One reason for this is that the above equation is the Euler equation of the nonconvex functional and criticality alone is not enough to prove the optimal L ∞ -bound for minimizers (see [4] and the references therein).On the other hand Theorem 1.1 could still hold without the monotonicity assumption on f .The proof of Theorem 1.1 is rather long and has several stages and therefore we give its outline here.In this paper we develop further the ideas introduced in [10].The main difficulty is to overcome the lack of regularity and growth condition on f , and to avoid the constant C to depend on f .In order to do this we will revisit the proof of the classical Harnack inequality by Di Benedetto-Trudinger [8] in order to have a more suitable and sharper version which allows us to treat the nonhomogeneous case.
To overcome the lack of regularity and to have the constant independent of f , we use the fact that the equation is in divergence form and integrate it locally over the level sets of the solution (as in [16,25]).This will give us precise local information how fast the level sets decay.Similar method has been used to study global regularity for solutions of elliptic equations e.g. in [5].Here we use it to prove local estimates.
The first observation is that we have a good local estimate on the decay rate of the level set {u ≥ t} when we are in a ball B(x, r) whose radius is small r ≃ t/ F (t) and the density is not close to one (Lemma 3.3).The second observation is a measure theoretical lemma (Lemma 3.4) which states that the measure of the set where the density σ t is between 1/5 and 4/5 (which can be thought to be the "boundary" of the level set {u ≥ t}) is related to the measure of the set where the density is larger than 4/5 (which corresponds to the "interior" of the level set {u ≥ t}).This is of course very much related to the isoperimetric inequality.We combine these two lemmas and obtain the following sharp estimate (see Proposition (3.5)) for the decay rate of the level set for almost every t > 0 for which the level set has small measure, µ(t) ≤ |B 2 |/2.A similar estimate holds for η(t) = |{u < t} ∩ B 2 | when η(t) is small.Note that this estimate has two parts on the right hand side.The first one is the nonhomogeneous estimate and the second is the homogeneous one.In the sublinear case f (t) ≤ t we may integrate the above inequality and conclude that the solution is L ε -intergable.In the nonhomogeneous case the inequality may oscillate between the two estimates.Another issue in the proof is to overcome the fact that we do not have any growth condition on f .In [10] it was assumed that the nonhomogeneity is of type g(t)t, where g is slowly increasing function.The proof was based on the idea that under this assumption any unbounded supersolution blows up as the fundamental solution of the linear equation.This is certainly not true in our case.To solve this problem we study more closely subsolutions of (1.1) and prove an estimate (Lemma 4.3) which tells us that if f grows faster than linear function then a solution of (1.1) grows faster than a solution of the linear equation.We iterate Lemma 4.3 and obtain the following lower bound for subsolutions.Similar result is obtained also in [13].
loc (B(x 0 , 2R)) be a nonnegative subsolution of div(A(x)Du) ≥ f (u) for a constant c > 0 which is independent of u and f .
In fact, we will not need this result in the proof of Theorem 1.1 but only Lemma 4.3.However since Theorem 1.2 follows rather easily from Lemma 4.3 we choose to state it.At the end of Section 4 in Corollary 4.4 we show that Theorem 1.2 implies the Keller-Osserman condition for entire solutions of (1.1).This result has been proved in [13].
Let us briefly give a rough version of the proof of Theorem 1.1.We integrate (1.4) (and its counterpart for η) and conclude that there exists t ε > 0 such that µ(t ε ) ≤ ε and For simplicity assume that for every t > t ε we have the nonhomogeneous estimate in (1.4).Then integrating (1.4) gives Finally we conclude that it has to hold M = max B 1 u ≤ 4t ε .Indeed, otherwise Theorem 1.2 would imply for which is a contradiction when ε is small.The paper is organized as follows.In the next section we recall basic results from measure theory.In Section 3 we prove estimates for nonnegative supersolutions of (1.1).The main result of that section is Proposition 3.5.In Section 4 we prove estimates for subsolutions of (1.1) and prove Theorem 1.2 and show in Corollary 4.4 how it implies the Keller-Osserman condition for entire solutions.Finally in Section 5 we give the proof of Theorem 1.1.

Acknowledgment
This work was supported by the Academy of Finland grant 268393.

Preliminaries
Throughout the paper we denote by B(x, r) an open ball centred at x with radius r.In the case x = 0 we simply write Here P (E, U ) is the perimeter of E in U .Sometimes we call E a set of finite perimeter if it is clear from the context which is the reference domain U .Let E be a set of finite perimeter in U .The reduced boundary of E is denoted by ∂ * E. It is smaller than the topological boundary which we denote by ∂E.For any open set V ⊂ U it holds P (E, V ) = ´∂ * E∩V dH n−1 , where H n−1 is the standard (n − 1)-dimensional Hausdorff measure.Moreover the Gauss-Green formula holds Here ν is the outer unit normal of E which exists on ∂ * E. For an introduction to the theory of sets of finite perimeter we refer to [2] and [15].All the following results can be found in these books.
The most important result from geometric measure theory for us is the relative isoperimetric inequality.It states that for every set of finite perimeter E in the ball B R the inequality holds for a constant c which depends on the dimension.The proof of the main result is mostly based on this inequality.
We recall the coarea formula for Lipschitz functions.Let g be Lipschitz continuous in an open set U and let h ∈ L 1 (U ) be nonnegative.Then it holds The formula still holds if g is locally Lipschitz continuous and h|Dg| ∈ L 1 (U ).From Coarea formula one deduces immediately that almost every level set {g > t} of a Lipschitz function is a set of finite perimeter.In the case g ∈ C ∞ (U ) the level sets are even more regular, since by the Morse-Sard Lemma the image of the critical set In particular, almost every level set of a smooth function has smooth boundary.Let u ∈ L 1 (B 2 ) be a Borel measurable function.Throughout the paper we denote the sublevel sets by E t := {x ∈ B 2 : u(x) < t} and the superlevel sets by In Section 3 we estimate the differential of µ(t) (and η(t)) when u is nonnegative supersolution of the equation (1.1).The differential of µ(•) is a measure and we denote its absolutely continuous part by µ ′ .To avoid pathological situations (see [1]) we regularize the equation in order to work with smooth functions.Then by the result from [1] we may write µ ′ as for almost every t.
Let us turn our attention to elliptic equations in divergence form.As an introduction to the topic we refer to [9].
Finally u ∈ W 1,2 (U ) is a weak solution of (1.1) in U if it is both super-and subsolution.
As we already mentioned we will regularize the equation (1.1) and work with solutions of where A ε is smooth and elliptic and f ε is positive, nondecreasing and smooth.Moreover Let us denote the sequence of solutions by (u ε ) which have the same boundary values as the original solution u, i.e., u ε − u ∈ W 1,2 0 (B 2 ).Since (u ε ) are locally uniformly Hölder continuous we have that by monotone convergence.Hence, we may assume that the solution u in Theorem 1.1 is smooth, i.e., u ∈ C ∞ (B 2 ).Finally for De Giorgi iteration we recall the following lemma which can be found e.g. in [9, Lemma 7.1].Lemma 2.2.Let (x i ) be a sequence of positive numbers such that

Estimates for supersolutions
In this section we study nonnegative supersolutions of (1.1).We begin by proving Caccioppoli inequality.We also prove a version of the Caccioppoli inequality which involves the boundary term.Recall that Then for r < R it holds for a constant depending on λ and Λ. Moreover for almost every t > 0 we have Proof.Let us fix t > 0 and choose a testfunction ϕ(x) R−r .We get after using the ellipticity and Young's inequality that ˆEt∩Br The standard Caccioppoli inequality follows from Here χ E denotes the characteristic function of a set E.
For the second statement we integrate the equation (3.1) over the set E t ∩ B ρ and get for almost every t > 0. In the above formula one has to consider the correct representative of A(x) in order the left hand side to be measurable.For a rigorous argument see [25] and note that for almost every t.By the ellipticity we have Choose ρ = r+R 2 and integrate the previous inequality with respect to ρ over (r, ρ) and get By the previous Caccioppoli inequality we have Arguing as in (3.2) we get Therefore from (3.3) we get by Young's inequality Next we observe that when the measure of the sublevel set |E t ∩ B 2 | is very small compared to the ratio t/ F (t), we have the same estimates as in the linear theory.The next result follows from the previous Caccioppoli inequality together with the standard De Giorgi iteration.The argument is almost exactly the same as [9,Lemma 7.4] and therefore we write only the outline of the proof.
Then there is δ 0 > 0 such that if for some t > 0 it holds Proof.Let t > 0 satisfy the assumption of the lemma.For 0 By possibly decreasing the value of δ 0 we may assume that Using the Caccioppoli inequality (Lemma 3.1) and arguing as in [9, Lemma 7.4] we conclude that for every 1 ≤ ρ < R ≤ 2 and 0 < h < k ≤ t it holds Define We choose δ 0 > 0 such that δ 0 ≤ C −n 0 4 −n 2 .By the assumption on t we have and we conclude from Lemma 2.2 that lim i→∞ x i = 0.In other words inf We turn our attention to Proposition 3.5 which is the main result of this section.To that aim we will need two lemmas.In the first lemma we use the Caccioppoli inequality to study the local decay rate of level sets.Due to the nonhomogeneity of the equation we do this only in small balls B(x, r) with radius r ≤ t/ F (t). Due to the relative isoperimetric inequality the estimate depends whether the density |E t ∩ B(x, r)|/|B r | is close to one or close to zero.
To avoid problems we prefer to work with smooth functions.
Then for almost every t > 0 it holds ˆ{u=t}∩B(x 0 ,2r) Proof.Without loss of generality we may assume that x 0 = 0. We may also assume that Let us fix x i ∈ E t ∩ B r .For x i we define radius R i such that Therefore we deduce from the assumption Thus we conclude that R i ≤ r.Hence we obtain a family of balls B(x i , R i ) which cover E t ∩ B r and satisfy 0 < R i ≤ r.By the Besikovitch covering theorem [15, Corollary 5.2] we may choose a countable disjoint subfamily, say F, such that ξ(n Moreover by the definition of R i it holds where the last inequality follows from R i ≤ r ≤ t/ F (t). On the other hand the relative isoperimetric inequality yields Therefore (3.4) and (3.5) give for every ball B(x i , R i ) ⊂ B 2r in the cover.Summing (3.6) gives that and the claim follows.
Let us assume next We use Lemma 3.1 in B r and argue as above to conclude that where we used the fact that r ≤ t/ F (t).By the relative isoperimetric inequality we have as before Hence the claim follows.
Next we prove a measure theoretical lemma which is related to the relative isoperimetric inequality.To this aim we denote for every δ ∈ (0, 1] the truncated distance function to the boundary ∂B 2 by where c ∈ (0, 1] is a number which we will choose later.For every measurerable set A ⊂ B 2 we define density function Although it is not apparent from the notation, the function σ A depends on δ.The point of the following lemma is to study the size of the set where the density function of a given set A takes values between 1/5 and 4/5.Heuristically one may think that the set {σ A = 1/2} corresponds to the reduced boundary of A and that {1/5 < σ A < 4/5} forms a layer around it.The thickness of this layer depends on the Lipschitz constant of σ A which can be estimated by a simple geometrical argument as follows.Fix x ∈ B 2 and a unit vector e.When h > 0 is small we may estimate where C depends on the dimension.Therefore σ A is locally Lipschitz continuous and its gradient can be estimated by be the truncated distance (3.7) and let σ A be the density function (3.8).There is a constant c n which depends on the dimension such that Note that if A is a smooth set then by letting δ → 0 the previous lemma reduces to the relative isoperimetric inequality.
Proof.Let us fix δ ∈ (0, 1].Throughout the proof we denote for every s ∈ [0, 1] the superlevel sets of σ A by We may write σ A as a convolution Therefore by Fubini's theorem we have Hence it holds Since |A| ≤ We divide the proof in two cases.Let us first assume that there exists ŝ ∈ (7/10, 4/5) such that part of the level set A ŝ = {σ A > ŝ} is away from the boundary ∂B 2 , (3.12) We observe that for every x ∈ B Rn it holds d δ (x) ≥ cδ for a dimensional constant c > 0. By possibly decreasing δ we deduce form (3.10) that |A s ∩ B Rn | ≤ 6 7 |B Rn | for every 3/5 ≤ s ≤ 4/5.Then it follows from (3.9), from the coarea formula and from the relative isoperimetric inequality that Hence the claim follows from (3.12) since (3.15) Choose a vector field X(x) = (|x| − 2) x |x| and observe that by the definition of δ n (3.11) it holds for every |x| ≥ 2 − δ n .Therefore the Gauss-Green formula and (3.15) yield The inequality (3.14) then follows from Finally we use (3.9), the coarea formula and (3.14) to conclude We use the two previous lemmas to estimate the decay rate of the level sets.
and for almost every t > 0 with η(t) Here µ ′ is the absolutely continuous part of the differential of µ.
Proof.We will only prove the first inequality since the second follows from a similar argument.By Morse-Sard Lemma for almost every t > 0 it holds that |Du| > 0 on {u = t} ∩ B 2 and by [1] we may write Let us fix t > 0 for which this holds.Let us choose define the truncated distance by and denote the density function by Let us divide the proof in two cases and assume first that a large amount of the level set A t has low density, In particular, it holds For every x i ∈ A t ∩ {σ At (x) ≤ 4/5} we choose a ball B(x i , d δ (x i )) and thus obtain a covering of A t ∩ {σ At ≤ 4/5}.By the Besicovitch covering theorem [15, Corollary 5.2] we may choose a countable disjoint subfamily, say F, such that We observe that if B(x i , d δ (x i )) and B(x j , d δ (x j )) are two balls from F such that the enlarged balls B(x i , 2d δ (x i )) and B(x j , 2d δ (x j )) overlap each other, then their radii are comparable, Therefore since the balls in F are disjoint it follows that the enlarged balls B(x i , 2d δ (x i )) intersect every point in B 2 only finitely many times, say, η(n) times.In particular we have, .
Summing the above inequality and using (3.16) and (3.17) yield In other words −µ ′ (t) ≥ c t µ(t) and the claim follows in this case.Let us next assume that large amount of the level set A t has density close to one, (3.18) {σ In this case we do not cover the set {σ At > 4/5} but only the part where the density σ At is between 1/5 and 4/5.Then we estimate the measure of this set by Lemma 3.4 .
For every x i ∈ {1/5 < σ At ≤ 4/5} we choose a ball B(x i , d δ (x i )) and thus obtain a covering of {1/5 < σ At ≤ 4/5}.By the Besicovitch covering theorem we may choose a countable disjoint subfamily F such that Moreover as in (3.17) we have .
Summing the above inequality and using (3.19) and (3.20) yield Finally we use Lemma 3.4 to conclude that Therefore by the two previous inequalities and by (3.18) we get The result follows from δ = t F (t) .

Estimates for subsolutions
In this section we prove estimates for subsolutions of the equation (1.1).The most important result of this section for Theorem 1.1 is Lemma 4.3.Similar estimate is proved in [12] but we need the estimate from Lemma 4.3 exactly as it is stated.The proof is fairly standard.Instead of using capacity argument [12] we give a short proof based on De Giorgi iteration.After Lemma 4.3 we give the proof of Theorem 1.2 and show in Corollary 4.4 how it implies the Keller-Osserman condition.
We begin by recalling the following standard estimate for subsolutions of the linear equation (4.1) div(A(x)Du) = 0.
There is Proof.Let us recall that by De Giorgi theorem any subsolution v of (4.1) is locally bounded and satisfies sup The result follows by applying this to v = (u − u(x 0 )/2) + which is a subsolution of (4.1).
We recall the notation A t = {u ≥ t}.We have the following Caccioppoli inequality for subsolutions of (1.1).Lemma 4.2.Let u ∈ W 1,2 (B(x 0 , 2r)) be a subsolution of and denote M r := sup B(x 0 ,r) u.Then for every ρ < r and t < M r it holds Proof.Without loss of generality we may assume that x 0 = 0. We use testfunction ϕ Since f is nondecreasing we have for every x ∈ A t ∩ B r that Therefore we have ˆBr We denote w = (u − t) + ζ ∈ W 1,2 0 (B r ) and obtain by Young's and Sobolev inequalities ˆBr .
The result follows from Jensen's inequality ˆBρ (u − t) By the standard De Giorgi iteration we obtain the estimate we need.
Proof.By rescaling and translating we may assume that r = 1 and u is a subsolution of We need to show that Λ 0 ≥ c > 0.
Let τ, ρ be such that 1/2 < ρ < τ < 1 and h, k such that u(0)/2 ≤ h < k ≤ u(0).Then it follows from the assumption M 1 ≤ 2u(0) and Lemma 4.2 that Since M 1 ≤ 2u(0) we obtain Indeed, we argue by contradiction and assume that (4.3) does not hold.This implies It follows from Lemma 2.2 that lim i→∞ x i = 0.This means that sup which is a contradiction.Therefore we have (4.3) and the claim follows.
Proof of Theorem 1.2.Without loss of generality we may assume that x 0 = 0. Let us denote M 0 = u(0) and R 0 = 0. We define radii R k ∈ (0, R], where k = 1, . . ., K, such that the corresponding maxima For notational reasons we also define Note that because of the maximum principle the maximum is attained on the boundary.Since sup Therefore by the monotonicity of F it holds ˆMk−2 Summing the above inequality over k = 1, . . ., K yields

t) .
The result follows from the previous two inequalities.
Let us briefly discuss about the previous theorem.At the first glance the statement of Theorem 1.2 might seem unsatisfactory.First, one could think that the assumption u(x 0 ) > 0 in Theorem 1.2 is unnecessary.However, a simple example shows that it can not be removed.Similar example shows that one can not reduce the interval of integration on the left hand side from [m/4, M ] to [m, M ], i.e., the estimate is not true.To see this choose f such that f (t) = 0 for t ∈ [0, 1] and f (t) = 1 for t > 1. Construct a one dimensional solution of u ′′ = f (u) in (0, 1) by u(x) = 1 for 0 < x ≤ 1 − ε and u(x) = 1 2 (x − 1 + ε) 2 + 1 for 1 − ε < x < 1.This solution does not satisfy the estimate (4.4).
Corollary 4.4.Suppose that there exists a nonconstant subsolution u Proof.Since u is not constant there exists a point x 0 such that u(x 0 ) > inf B(x 0 ,1) u.Without loss of generality we may assume that x 0 = 0. Let us fix a radius R > 1 and denote the result follows by letting R → ∞.

Proof of the Main Theorem
This section is devoted to the proof of Theorem 1.1.where M = sup B 1 u.To that aim let ε > 0 be a small number which we will choose later.Let t ε ≥ t 0 be the first value of u such that or, to be more precise, t ε := sup{t > 0 : µ(t) > ε}.
Lemma 3.5 implies that for every t ∈ (t 0 , t ε ) it holds However since M k ≥ sup B(x k−1 ,r k ) u this contradicts the fact that M k was chosen such that M k = 2M k−1 .Hence we have (5.5).Note also that it follows from 0 < sup B(x k−1 ,r k ) u ≤ 2u(x k−1 ) = 2M k−1 and from Lemma 4.3 that when n ≥ 2. If Thereom 1.1 would be true for nonnegative solutions of (5.8) then there would be C such that (5.9) ˆM m 1 Let ϕ ≥ 0 be fundamental solution of the Laplace equation with inf B 1 ϕ = 1 and singularity at the origin.Since ϕ is unbounded we find a radius r > 0 such that for the value of ϕ on ∂B r , denote it by T , it holds (5.10) We define u ∈ C 1,1 (B 2 ) such that u(x) = ϕ(x) for x ∈ B 2 \ Br and u(x) = a(r 2 − |x| 2 ) + T for x ∈ B r .Here a > 0 is chosen such that u ∈ C 1,1 (B 2 ).Then u is a solution of (5.8) for some f which satisfies f (t) = 0 for t ∈ (0, T ).Therefore (5.9) can not hold because of (5.10).
for a constant c which depends on ε.In other words