Positive evanescent solutions of nonlinear elliptic equations ...

J. Math. Anal. Appl. 333 (2007) 863–870 www.elsevier.com/locate/jmaa

Positive evanescent solutions of nonlinear elliptic equations Smail Djebali a , Toufik Moussaoui a , Octavian G. Mustafa b,∗ a Department of Mathematics, ENS, PO Box 92, 16050 Kouba, Algiers, Algeria b Department of Mathematics, DAL, University of Craiova, Romania

Received 20 October 2005 Available online 9 January 2007 Submitted by H. Gaussier

Abstract In this note we investigate the existence of positive solutions vanishing at +∞ to the elliptic equation u + f (x, u) + g(|x|)x · ∇u = 0, |x| > A > 0, in Rn (n 3) under mild restrictions on the functions f , g. © 2006 Elsevier Inc. All rights reserved. Keywords: Positive solution; Nonlinear elliptic equation; Exterior domain

1. Introduction The elliptic equation u + f (x, u) + g |x| x · ∇u = 0,

x ∈ Rn , |x| > A > 0

(1)

(by |x| we mean the usual euclidean norm) encompassing numerous equations from mathematical physics, has been studied extensively in the last years. Constantin [3] established that (1) has positive solutions provided that f (x, u) 0 and g is bounded. In a further work [4], concerned with the existence of positive solutions to Eq. (1) that * Corresponding author. Correspondence address: Str. Tudor Vladimirescu, Nr. 26, 200534 Craiova, Jud. Dolj,

Romania. E-mail addresses: [email protected] (S. Djebali), [email protected] (T. Moussaoui), [email protected] (O.G. Mustafa). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.12.004

864

S. Djebali et al. / J. Math. Anal. Appl. 333 (2007) 863–870

vanish at infinity (a phenomenon called evanescence), the boundedness of g has been replaced with an integral restriction +∞ r g(r) dr < +∞.

(2)

A

Very recently, Ehrnström [7] noticed that (2) is unnecessary if g is assumed nonnegative. An inspection of the proof in [7] establishes that (2) can be replaced naturally by +∞ rg − (r) dr > −∞, A

where g − (r) = min(0, g(r)) for all r A. Similar investigations regarding (1) were performed in [5,10–12]. Throughout these papers different methods have been employed: the Banach contraction principle and exponentially rescaled metrics, subsolutions and supersolutions, variational techniques. Set GA = {x ∈ Rn : |x| > A}, n 3. Our general hypotheses are that f : GA × R → R is locally Hölder continuous and g : [A, +∞) → R is continuously differentiable, similarly to [9]. In the spirit of [4,7], we shall specialize the nonlinearity by asking that 0 f (x, u) Ma |x| u, x ∈ GA , u ∈ [0, ε], (3) for given M, ε > 0. Here, a : [A, +∞) → [0, +∞) is continuous. In what concerns the function g, we shall assume in the sequel that it takes only nonnegative values. To establish the existence of positive evanescent solutions to (1) in GB , for some B A, the recent paper [7] relies on the condition +∞ ra(r) dr < +∞.

(4)

A

An even stronger condition has been employed in [10] +∞ r n−1 a(r) dr < +∞. A

The aim of this note is to show that such solutions exist even if condition (4) does not hold. We mention at this point that our method seems unfitted to attack the case n = 2 that has been treated by Constantin [2] under stronger restrictions. The main result will be established using the subsolution and supersolution approach and the strong maximum principle. By a subsolution of (1) we understand a function w ∈ C 2 (GB ) ∩ C(GB ) such that w + f (x, w) + g(|x|)x · ∇w 0 for |x| > B. As for the supersolution, the sign of the inequality should be reversed. 2. The evanescent solution The following lemma will be needed in our investigation.


865

Lemma 1. [9] If, for some B > A, there exist a nonnegative subsolution w and a positive supersolution v to (1) in GB , such that w(x) v(x) for x ∈ GB , then (1) has a solution u in GB such that w u v throughout GB . In particular, u = v on |x| = B. To establish the main result, a lemma regarding the existence of a special solution to a linear ordinary differential equation of second order is essential. Lemma 2. Let c, ε > 0 be fixed. Let q : [A, +∞) → [0, +∞) be continuous and assume that it does not become identically zero on any subinterval of [A, +∞). Suppose further that there exist λ ∈ (0, 1] and S0 max(e2 , A) such that +∞ q(t) dt λ s log s

(5)

s

for all s S0 . Fix ε1 ∈ (0, 1 − λ) if λ < 1 and ε1 ∈ (0, 1) in the limiting case λ = 1. Then the equation h (s) + q(s)h(s) = 0,

s A,

(6)

has a solution h that satisfies the conditions below: (i) for all s S1 , where S1 S0 is sufficiently large, it holds that h(s) c,

h(s)

c1 + c , c1

log s c + c1 0. s

Given the choice of c1 , ε1 and taking into account (5), we also have

(10)


0 < h0 (s) λc1 + s log s

867

+∞ q(t) dt h0 (s) s

λh0 (s) h0 (s)ε1 λh0 (s) cε1 + + s log s s log s s log s s log s λ + ε1 h0 (s) 1 h0 (s) · < · , s S1 . log s s log s s We take x0 = T n (x0 ) in (10) to establish that lim sup sx0 (s) λn c1

(11)

s→+∞

for all n 1. By letting n → +∞ in (11) we get lims→+∞ sx0 (s) = 0 and, via L’Hôpital’s rule, s c + S1 x0 (t) dt h0 (s) = lim = 0. lim s→+∞ log s s→+∞ log s Case λ = 1. Take c1 ∈ (0, ε1 c), S1 S0 such that (9) holds and C, T : C → C as before. A partial order on C is given by the usual pointwise order “,” that is, we say that x1 x2 if and only if x1 (s) x2 (s) for all s S1 , where x1 , x2 ∈ C. Since q is nonnegative, the application T is isotone, that is, T x T y whenever x y, and it satisfies 0 T (0). By application of the Knaster–Tarski fixed point theorem [6, p. 14], T has a fixed point in C, denoted x0 . The proof follows the same lines as in the case λ ∈ (0, 1). 2 Remark 1. The function q with q(s) =

λ 1 , · 2 s log s

s max(e, A),

satisfies (5) but not the more restrictive condition +∞ sq(s) ds < +∞,

(12)

A

employed in [7] to get a solution h of (6) that verifies 0 h (s)
1 Eq. (14) does not have any solution satisfying either (7) or (8), we see that hypothesis (5) is optimal for encountering this type of asymptotic behavior at certain solutions of (6). Our main contribution here is the next result. Theorem 1. Let (3) hold and assume that there exist λ ∈ (0, 1], ε1 ∈ (0, 1 − λ sgn(1 − λ)), S0 max(e2 , A) such that for all s S0 at least one of the restrictions below holds true:

(i)

0 A,

(21)

has a positive evanescent solution under the hypotheses (3), (4), its “small” perturbation by the term “g(|x|)x · ∇u,” namely (1), where the degree of “smallness” is given by (2), will preserve this feature. In our case, however, it might happen that for certain functions a, g, where M a(s), s S0 , n−2 such that (17) holds the unperturbed equation (21) does not have any vanishing at +∞ solution besides the trivial solution. 0 g(s)

Remark 5. In [4,7] the decay of the positive evanescent solution of (1) is given by u(x) = O( |x|1n−2 ) as |x| → +∞ in contrast with the behavior u(x) = O( log(|x|) ) obtained here. |x|n−2 Acknowledgments This work has been completed during the visit of O.G.M. to the Mathematics Department of Lund University, Sweden, in August 2005 and has been supported by the National Science Foundation of Sweden through the grant VR 621-2003-5287. O.G.M. thanks the Mathematics Department for making his stay very agreeable. The authors are indebted to an anonymous referee for very helpful suggestions.

References [1] F.V. Atkinson, On second order nonlinear oscillation, Pacific Math. J. 5 (1955) 643–647. [2] A. Constantin, Positive solutions of elliptic equations in two-dimensional exterior domains, Port. Math. 52 (1995) 471–474. [3] A. Constantin, Existence of positive solutions of quasilinear elliptic equations, Bull. Austral. Math. Soc. 54 (1996) 147–154. [4] A. Constantin, Positive solutions of quasilinear elliptic equations, J. Math. Anal. Appl. 213 (1997) 334–339. [5] A. Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl. 184 (2005) 131–138. [6] J. Dugundji, A. Granas, Fixed Point Theory, vol. I, Polish Sci. Publ., Warszawa, 1982. [7] M. Ehrnström, Positive solutions for second-order nonlinear differential equations, Nonlinear Anal. 64 (2006) 1608– 1620. [8] L.E. Fraenkel, Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Univ. Press, Cambridge, 2000. [9] E.S. Noussair, C.A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains, J. Math. Anal. Appl. 75 (1980) 121–133. [10] A. Orpel, On the existence of positive radial solutions for a certain class of elliptic BVPs, J. Math. Anal. Appl. 299 (2004) 690–702. [11] E. Wahlén, Positive solutions of second-order differential equations, Nonlinear Anal. 58 (2004) 359–366. [12] Z. Yin, Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal. 54 (2003) 391–403.