POSITIVE SOLUTIONS TO ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER

. This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.


1.
Introduction. The structure of positive harmonic functions on a domain Ω in R N (N ≥ 2) has been much studied. Early in 1941 Martin [13] gave a method for uniquely representing any positive harmonic function in an arbitrary domain in R 3 by an integral on the minimal Martin boundary. His results have been extended to second order elliptic operators with a zero potential by Shur [16]. In the case where the closure of a domain is compact in the manifold, many mathematicians gave sufficient conditions for the corresponding Martin boundary to be equal to the relative boundary of the domain(see Hunt and Wheeden [9] and Taylor [17]). There are also some investigation fixed on the positive harmonic functions in some special unbounded domain, such as half-space, cone or cylinder. For example, Benedicks [1] has established a harmonic measure criterion that describes when the cone of positive harmonic functions on Ω that vanish on the boundary ∂Ω is generated by two linearly independent minimal harmonic functions. Benedicks' criterion describes when a Denjoy domain behaves like the union of two half-spaces from the point of view of potential theory. Related work, based on sectors, cones or cylinders, may be found in [3,12,7,15]. Landis and Nadirashvili [11] showed that a positive solution to a uniformly elliptic equation in a cone of R n which vanishes at the boundary is unique up to a constant multiple. Murata [14] established a method to construct the Martin boundary and Martin kernel for second order elliptic equations and gave a sufficient condition for an equation in R n or a cone of R n to have a unique (up to a constant multiple) positive solution vanishing at the boundary.
Assume C is the cylinder D × R, where D is a bounded Lipschitz domain in R n+1 , and x = (x 1 , · · · , x n+1 ) = (x 1 , · · · , x n , y) = (x , y) denote a typical point of R n × R. This paper investigates positive solutions of linear elliptic equation defined in C. When the cylinder is U = B × R (here B is the unit ball in R n ), it is known (see [6]) that the cone of positive harmonic functions h ± (x , y) = e ±αy φ(x ), where α is the square root of the first eigenvalue of the operator −∆ = − n j=1 ∂ 2 ∂x 2 j on B and φ is the corresponding eigenfunction, normalized by φ(0) = 1. We want to show the set of the positive solutions of elliptic equation (1) defined in C has a similar structure.
We consider the following elliptic equation: where L stands for second order uniformly elliptic operator of one of the following two types: We assume that a ij (x) = a ji (x) ∈ C ∞ (C), and L is uniformly elliptic, for some Λ > 0 and any ξ ∈ R n+1 .
The assumption a ij ∈ C ∞ is qualitative, in the sense that none of our estimates depend on the smoothness of a ij . By the standard approximation technique, all of our results are valid to uniformly elliptic equations with measurable coefficients a ij .
We are interested in the question of existence and uniqueness (to within a multiplicative constant) of a solution u of problem (1). We also study the precise asymptotic behaviors of the solutions and show finer properties of these solutions in the cylinder. In particular, there are two special solutions with exponential growth at one end while exponential decay at the other and all the positive solutions are linear combinations of these two.
Theorem 1.1. Solution set S + and S − of problem (1) are not empty. S is a linear combination of S + and S − . That is, for any v ∈ S + , w ∈ S − , we have The asymptotic behaviors of the solutions: There exist constants α, β, C, C > 0 depending only on n, Λ, D, such that, for any v ∈ S + \ {0}, w ∈ S − \ {0}, For any u ∈ S ∨ . We assume x * = (x * , y * ) ∈ C, such that u(x * ) = m(u). Then In order to illustrate the results, we give a simple example which is also a special case of Therem 1 in Gardiner [6].
Define S F is the solution set of problem (6). It is easy to see that e y sin x, e −y sin x ∈ S F . Actually they are the only two nontrivial solutions in S F in the sense of Theorem 1.1. We also see that the asymptotic behavior of solutions is exponential.
Particularly, for any y ∈ R, we use C y := C {y} , C + y := C (y,+∞) , C − y := C (−∞,y) , C + := C + 0 , C − := C − 0 . We also study the positive bounded solutions defined in half cylinder C + . They can be approximated by the solution in S − . That is Then for w ∈ S − , there exist constants α > 0, which is only dependent of n, Λ, D, and K, C > 0 which are only dependent ofû(1), n, Λ, D, such that The rest of this paper is divided into two parts. In Section 2 we establish some auxiliary results. First we introduce some fundamental notions concerning the positive solutions to equation (1) vanishing at the boundary. The maximum principle for the solutions in cylinder is proved under a bounded condition. We then introduce a so-called boundary Harnack principle which is proved in Fabes et al. [5] The main theorem will be proved in Section 3. We study the structure of the set of the positive solutions. We will show the exponential growth and decay for the positive solutions. The existence of the positive solutions is also proved. Any bounded positive solution in half cylinder can be approximated by the solution in the whole cylinder.

2.
Preliminaries. In this section, we collect some preliminary results. In Section 2.1, we shall give the maximum principle in cylinder. In Section 2.2, on the basis of the boundary Harnack principle, we give some lemmas to compare the solutions.
2.1. Maximum Principle. According to the well-known Maximum Principle a subharmonic up-bounded continuous function defined in a domain Ω ⊂ R n , n ≥ 1 which is non-positive on the boundary ∂Ω, is in fact negative everywhere in Ω and this result extends to nonnegative solutions of a large class of linear elliptic equation [8]. It is an important tool for us to study the properties of the solutions. Now we want to investigate the validity of the maximum principle for the solutions in cylinder. We begin our investigation with a Harnack-type principle for operator L.
By classical maximum principle, u(x) ≤ u + (x) ≤ 1, x ∈ C (0,2) . With Boundary Hölder estimate in Gilbarg and Trudinger [8] (Theorem 6.19 in non-divergence form and Theorem 8.25 in divergence form), there exists a constant C 0 > 0 which is . Take ε 0 sufficiently small such that C 0 ε α 0 ≤ 1 2 and apply Harnack principle When L is in non-divergence form in Lemma 2.1, there is an alternate proof using barrier function. As a matter of fact, assume D ⊂ B R ⊂ R n , set Now we can extend the maximum principle to the solutions in cylinder C.
Lemma 2.2. Suppose Lu(x) ≤ 0, x ∈ C, and u is bounded from above. Then Proof. Without loss of generality, we assume sup Set The solutions defined in half cylinder C + can be proved to satisfy the following maximum principle through the same method.
Now we are able to show the exponential decay of the bounded solutions in C + .
Proof. By the definition of m(u) andû(y), we assume the minimize sequence We claim that there exists a subsequence of {(x j , y j )} j (we still denote it {(x j , y j )} j ), such that lim j→∞ y j = −∞ or lim j→∞ y j = +∞.
We claim there exists a constant a > 0, such that av(x) ≤ u(x), x ∈ C. By contradiction, suppose there exist {x j = (x j , y j )} +∞ j=1 ⊂ C, such that 1 j v(x j ) > u(x j ). Then there is a subsequence of {x j } +∞ j=1 (still denote it {x j } +∞ j=1 ), such that lim j→∞ y j = +∞. It is easy to see there is a lower bound for {y j } +∞ j=1 from (9). Actually if {y j } +∞ j=1 is bounded in R, we assume that |y j | < M, j ≥ 1.
With Lemma 2.8, there exists a constant k M > 0, such that If we take J ∈ N + sufficiently large such that 1 We get a contradiction with (10).
Without loss of generality, we assume that {y j } +∞ j=1 is a strictly monotone increasing sequence, that is y j2 > y j1 > 0, j 2 > j 1 ≥ 1(we can subtract subsequence to assure it). Assume ε is the constant in Lemma 2.8. For any j ≥ 1, we consider u(x) and 1 j u(x) in area C (0,yj ) . There is a point With lim j→∞ y j = +∞, From Proposition 1, u(x) is not bounded in the area C (0,+∞) , and we get a contradiction. So there is a positive constant a > 0, such that av ≤ u. With the similar method, we can show there is a positive constant b > 0, such that bw ≤ u.
Proposition 3. For any u, v ∈ S + , there exists a constant c > 0, such that Proof. With Lemma 2.8, for any y ∈ R, there exists a constant c y , such that We prove (11) by contradiction. Suppose there exist {x j = (x j , y j )} +∞ j=1 ⊂ C, such that u(x j ) > jv(x j ). With Lemma 2.8, there exist a constant ε, such that u(x , y j ) ≥ εjv(x , y j ), x ∈ D. If we choice j = N big enough such that εN > c 0 +1, then this contradicts with (12).
3. Proof of the main theorem. With the preparations in above section, we will finish the proof of the main theorems in the section.
Proof of Theorem 1.1. We divide the proof into three parts. 1 o First we study the structure of S + and S − . For any u, v ∈ S + , set By contradiction. If Kv − u = 0, then Kv − u ∈ S + . With Proposition 3, there exists a constant K 1 > 0, such that v(x) ≤ K 1 (Kv(x) − v(x)), and Then K − 1 K1 ∈ E, and this contradicts with the definition of K. Therefore we get u = Kv, and S + = {av|a > 0}. With the similar method we can also get S − = {bw|b > 0}.
2 o Next we study the structure of the solution set S.
For any u ∈ S, if u ∈ S + or u ∈ S − , then there is a > 0 or b > 0 , such that u = av or u = bw. Next we suppose u ∈ S ∨ . Assume With Proposition 2, E = ∅, so a * > 0. We also have a * < +∞. Actually if not we can follow the method in Proposition 3 to get a contradiction.
Consider the function u − a * v. By the continuity, From u / ∈ S + and Harnack principle, For u − a * v ∈ S, we claim that u − a * v ∈ S − . Then there exists b > 0, such that u − a * v = bw. It is easy to see that So we get the conclusion. We prove the claim by contradiction, if u − a * v ∈ S ∨ , then by Proposition 2, there is a constantā > 0, such that u − a * v ≥āv, u − (a * +ā)v ≥ 0. So a * +ā ∈ E, this contradicts with a * = sup E.
3 o The existence of the solutions: For H ∈ N + , consider the equation With the classical elliptic theory, there exists a unique positive solution u H (x) to equation (13) and u H (x) > 0, x ∈ C (−H,+H) . For (13) is linear, we can adjust the constant C, such that u H (0 , 0) = 1.   1] . When M = 2, we can also get a subsequence of {u 2] . We do the similar operations when M ∈ N + , and get a sequence {u  M ] , M ∈ N + . We denote the limit by u(x) which is defined in C. So u(x) satisfies (1). Therefore u(x) ∈ S. Moreover we can prove that u(x) ∈ S + .
With the similar method we can prove that there exists w(x) ∈ S − . It is easy to check (u + w)(x) ∈ S ∨ = ∅.
Proof of Theorem 1.2. We divide the proof into two steps. In Step 1, we prove that the growing and decaying rate of positive solutions to (1) is at least exponential in the infinity. In Step 2, we prove the rate of asymptotic behaviors of these positive solutions is at most exponential in the infinity.
Therefore we get a part of the inequality in (3). With the similar method we get a part of the inequality in (4).
With a similar argue we get the estimate in (−∞, y * ). Therefore we get the rest of (5).
We use Lemma 2.3 and get If (20) is satisfied, with a similar argue we also get (21).