Existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends

Nonlinear Analysis 55 (2003) 83 – 101

www.elsevier.com/locate/na

Existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends Kazuhiro Kurataa;∗ , Masataka Shibatab , Kazuo Tadac a Department

of Mathematics, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachiouji-shi, Tokyo 192-0397, Japan b Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan c 1-212-62-1 Shimokiyoto, Kiyose-shi, Tokyo 204-0011, Japan Received 23 April 2003; accepted 6 June 2003

Abstract In this paper, we study the existence of positive solutions to nonlinear elliptic boundary value problems on unbounded domains ! ⊂ Rn with cylindrical ends for a general nonlinear term p f(u) including f(u) = u+ ; 1 ¡ p ¡ (n + 2)=(n − 2)(n ¿ 3); + ∞ (n = 2) as a typical example: −5u + u = f(u); u ¿ 0 (x ∈ !); |@! = 0; u(x) → 0 (|x| → ∞) by using the mountain pass approach. The geometry of ! plays an important role in our analysis. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Nonlinear elliptic equation; Variational problem; Least energy solution; Cylindrical unbounded domains

1. Introduction We consider the nonlinear elliptic boundary value problem − 5u + u = up ;

u¿0

(x ∈ );

u|@ = 0;

u(x) → 0

(|x| → ∞);

(1)

where is an unbounded domain in Rn with the boundary @ of locally piecewise C 1 class, 1 ¡ p ¡ (n + 2)=(n − 2) (n ¿ 3), + ∞ (n = 2), and is a parameter. We assume ¿ 0 throughout this paper, for simplicity, although one can allow to be negative ∗

Corresponding author. Tel.: +81-426-77-2459; fax:+81-426-77-2481. E-mail addresses: [email protected] (K. Kurata), [email protected] (M. Shibata), [email protected] (K. Tada). 0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00214-1

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to some extent for domains in which PoincarCe’s inequality holds. In 1982, Esteban and Lions [13] discovered a certain criterion of unbounded domains in which the BVP above has no solution. For example, there exist no solution for the semi-inInite cylindrical domain

= (0; + ∞) × !; where ! ⊂ Rn−1 is a bounded domain. Actually, they proved non-existence of nontrivial energy Inite solution to (2), if there exists a constant vector X ∈ Rn such (x) · X ¿ 0 and (x) · X ≡ 0 for x ∈ @ , where (x) is the outward unit normal vector at x ∈ @ . On the other hand, in 1983, several peoples (e.g., [12,2,23]) proved the existence of a solution on the inInite (straight) cylindrical domain =(−∞; + ∞) × !. After that, in 1993 Lien et al. [16] proved the existence of a solution on unbounded domains with a periodic structure and their locally deformed domains, precisely adding bounded domains, by using concentration-compactness principles. They also proved the existence of a solution on a domain

= {x ∈ Rn ; |x| ¡ R} ∪ {x = (x1 ; x ) ∈ R × Rn−1 ; x1 ∈ (0; + ∞); |x | ¡ r} for Ixed r ¿ 0 and suKciently large R ¿ r. Also, del Pino and Felmer [10] proved similar results, but in slightly diLerent situations, for more general nonlinearity by using the mountain pass approach. We also note that Bahri and Lions [3] have proved the existence of a solution on any exterior domain for ¿ 0. For an existence result on Rn \C, where C is an inInite cylinder or an inInite cylindric annulus, we refer the work of Dall’aglio and Grossi [8]. However, the relation between the shape of an unbounded domain and the solvability of the BVP (1) is still unclear. In this paper, we propose a class of unbounded domains, domains with semi-inInite cylindrical ends (the precise deInition is given in Section 2), in which the BVP will be solved. Actually we present two conjectures on the solvability of the BVP (1) and give several results to support these conjectures. In Section 2, we consider the elliptic boundary value problem with a general nonlinp earity f(u), including f(u) = u+ as a special case. We introduce a class of unbounded domains with semi-inInite cylindrical ends and give two conjectures on the solvability on such domains. We state two results (Theorems 1 and 2) on the existence of a least-energy solution of the mountain pass type (we just call it a least-energy solution) and a result (Theorem 4) on the existence of a higher energy solution. We also give some symmetry properties (Theorem 3) of a least-energy solutions on domains with symmetries with respect to axes. In Section 3, we give the proof of Theorems 1, 2 and 3. In Section 4, we give the outline of the proof of Theorem 4. 2. Main results We consider the nonlinear elliptic boundary value problem with a general nonlinear term f(u) −5u + u = f(u); u(x) → 0;

u¿0

(|x| → ∞);

(x ∈ );

u|@ = 0; (2)

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85

where is an unbounded domain in Rn with the boundary @ of locally piecewise C 1 class and ¿ 0 is a parameter. Here, f(t) is a C 1 function satisfying the following conditions: (f-1) f(t) = 0 for t 6 0 and f(t) = o(t) as t → 0; (f-2) there exists p ¿ 1 such that p ¡ (n + 2)=(n − 2) for n ¿ 3 and p ¡ + ∞ for n = 2 and f(t) lim = 0; t → + ∞ tp (f-3) there exists  ∈ (2; p + 1] such that 0 ¡ F(t) 6 f(t)t

for t ¿ 0;

(f-4) the function t →
f(t)=t is strictly increasing on (0; + ∞), t where F(t) = 0 f(s) ds. To introduce the class of unbounded domains to be considered in this paper, we denote by S(!) and A(!) the inInite cylinder and the semi-inInite cylinder, respectively, with a bounded domain ! ⊂ Rn−1 as its cross-section S(!) = {x = (x1 ; x ) ∈ R × Rn−1 ; x1 ∈ (−∞; + ∞); x ∈ !}; A(!) = {x = (x1 ; x ) ∈ R × Rn−1 ; x1 ∈ (0; + ∞); x ∈ !}: Especially, we use the notation SR = S(B (O; R)) and AR = A(B (O; R)) for B (O; R) = {x ∈ Rn−1 ; |x | ¡ R} with R ¿ 0. 

Denition 1. If there exist m ∈ N, a bounded domain ! ⊂ Rn−1 and a compact set K such m 

∩ Kc = A(j) (!); j=1

where each A(j) (!) is congruent with A(!), then we say that is a domain with m semi-inInite cylinder A(!) as its ends (see Fig. 1). From SR we construct the V -shaped cylindrical domain, we denote by SR(V ) (see Fig. 2), by the following procedure: cutting the domain SR via a hyperplane, not parallel to the cross-section, and attaching again its new cross-sections so that points on one cross-section are transformed into the points of the other cross-section, which is symmetric with respect to its center. We can continue this procedure to construct a Initely times bent domain from SR . One can also consider the smoothly locally bent cylindrical domain with a ball of same radius R as its cross-section everywhere: Let y = y(s); s ∈ (−∞; + ∞); be a smooth curve in Rn which is a straight line outside a compact set and let P(s) be a set of unit vectors which are perpendicular to the tangent vector y (s). Then such domain can be described as follows:

= {x = y(s) + t(s); s ∈ (−∞; + ∞); (s) ∈ P(s); t ∈ [0; R)}:

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K. Kurata et al. / Nonlinear Analysis 55 (2003) 83 – 101

Fig. 1. A domain with four semi-inInite cylinder AR as its ends.

2R

(V )

Fig. 2. V-shaped cylindrical domain SR .

We conjecture the following two statements for the solvability of (2) on a domain

with m semi-inInite cylinder A(!) as its ends. Conjecture 1. If is either a 7nitely times bent domain or a smoothly locally bent domain constructed from SR by the procedure above, then there exists a least-energy solution (the precise de7nition is given later) to (2). Actually, we conjecture the stronger statement c( ) ¡ c(SR ) for such domains (the de7nition of the mountain pass value c( ) is given later). In the proof of Theorem 1, one can see that once we know c( ) ¡ c(SR ) we can show the existence of a least-energy solution. This kind of criterion to recover compactness is well known in many problems (see, e.g., [4,17]). The diKculty here is to understand how the geometry of aLects the validity of the strict inequality c( ) ¡ c(SR ). When has the geometry above, it is known in the quantum wave guide theory (see, e.g., [10]) that the ground energy ∇u2L2 ( ) E( ) = inf 2 u∈H01 ( );u=0 uL2 ( ) satisIes E( ) ¡ E(SR ). We are also motivated to show a nonlinear version of such interesting phenomena in Conjecture 1.

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Conjecture 2. If m ¿ 2 and is a domain with m semi-in7nite cylinder A(!) as its ends, then there exists a solution to (2). In general, we cannot expect the existence of a least-energy solution of mountain pass type to (2) under the situation of Conjecture 2. Since it is well known that under our assumptions on f(t) a solution obtained by a mountain pass approach has the least-energy among solutions to (2), we call it just a least-energy solution throughout this paper. For certain domains, we cannot expect the existence of the mountain pass solution and must ask the existence of solutions with a higher energy than the mountain pass value c( ). We also remark that PoincarCe’s inequality holds on unbounded domains with such cylindrical ends (see, e.g., [21]). To state our Irst result, we denote (V ) by SR; L the semi-in7nite V -shaped cylindrical domain which is constructed by cutting, perpendicularly by a hyperplane, an inInite part of one of the semi-inInite part of SR(V ) remaining a Inite part with length L, measured from certain point on the bent region. (V ) (V ) (V ) as L → ∞. We also use the same notation SR ; SR(V ) ; SR; So, SR; L for the L tends to SR (V ) (V ) domains which are congruent with SR ; SR ; SR; L , respectively. Now, we state our Irst result. Theorem 1. Suppose ¿ 0 and (f-1) – (f-4) for f(t). Let m ¿ 1, R ¿ 0 and be a domain with m semi-in7nite cylinder AR as its ends which satis7es the additional condition: (V ) (∗) contains one of SR , SR(V ) and SR; L with su:ciently large L ¿ 0. Then, there exists a least-energy solution to (2). For the case that contains SR , Theorem 1 has been proved essentially in [10,16]. The result for the case = SR(V ) seems new as far as we know and also plays an important role in the proof of Theorem 1 for the general case. As a special case, we consider the problem − 5u = up ;

u¿0

(x ∈ );

u|@ = 0;

u(x) → 0

(|x| → ∞);

(3)

where is an unbounded domain in Rn (n ¿ 2) and 1 ¡ p ¡ (n + 2)=(n − 2) for n ¿ 3 and 1 ¡ p ¡ + ∞ for n = 2. We obtain the following result to (3). Theorem 2. Let be a 7nitely bent domain constructed from SR , 7x its shape and consider R as a parameter. Then, there exists a su:ciently small R0 such that for every R ∈ (0; R0 ) there exists a least-energy solution to (3). We can show Theorem 2 by combining a similar argument as in the proof of Theorem 1 and the scaling argument. Remark 1. One can see in the proof of Theorems 1 and 2 that the statements are true even in the case that the cross-section B (O; R) is replaced by a bounded domain ! ∈ Rn−1 which is convex and symmetric with respect to axes xj ; j = 2; : : : ; n. Because we just use the existence and symmetry properties of a least-energy solution on S(!).

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Theorems 1 and 2 give partial answers to Conjecture 1. However, the complete answer to Conjecture 1 remains open even for Initely bent domains constructed from SR . Moreover, existence of the least-energy solution for a smoothly locally bent domain constructed from SR is also an open problem, although we can obtain the existence result for certain smooth domains close to V -shaped domain, which are constructed smoothing the corner of V -shaped domains slightly. Recently, Shibata [20] proved existence of positive solutions for a smoothly locally bent domain in two-dimensional case under certain smallness condition on the curvature or the width of the cylinder. Here, we briePy recall the mountain pass approach to Ind a solution to (2). Problem (2) has a variational structure and a solution u to (2) can be characterized as a non-trivial critical point of the energy functional   1 J (u) = (|∇u|2 + u2 ) d x − F(u) d x: (4) 2

Under assumptions (f-1) – (f-3), it is well known (see, e.g., [10,18,19] that J (u) has a mountain pass structure and, when (f-4) is assumed, its mountain pass value c( ) can be written by c( ) =

sup J ((u):

inf

u∈H01 ( );u+ ≡0 (¿0

It is also known that this characterization implies that c( ) =

inf

u∈M;u+ ≡0

J (u);

where M is a solution manifold (called the Nehari manifold), which includes any solution to (2): namely,     1 M = u ∈ H0 ( ); u(−5u + u) d x = f(u)u d x :



J  (u)

= 0, then u has the least-energy among This means that if J (u) = c( ) and any solutions to (2). So, we call u a least-energy solution to (2), if J (u) = c( ) and J  (u) = 0. Next, we remark on symmetry property of solutions to (2) on symmetric domains. When = SR , in [6] (see also [5]) they proved by the moving plane method that any solutions u to (2) has the symmetry u(x1 ; x ) = u(x1 ; |x |);

u(x1 ; x ) = u(−x1 ; x )

for any x = (x1 ; x ) ∈ SR . Especially, when n = 2, uniqueness of solutions to (2) is also p known by Dancer [9], at least for the case f(u) = u+ and = 0. Now, we consider (2) on the domain (see Fig. 3)

= {x = (x1 ; x2 ) ∈ R2 ; |x1 | 6 R=2 or |x2 | 6 R=2}: This domain has a symmetry with respect to the x1 -, x2 -, *-, and +-axis; * = {x; x1 = x2 };

+ = {x; x1 = −x2 }:

For this domain, it is easy to see that any solution to (2) is symmetric with respect to x1 - and x2 -axis by using the moving plane method as in [5,6]. For a least-energy

K. Kurata et al. / Nonlinear Analysis 55 (2003) 83 – 101

89

Fig. 3.

p ), we also show solution obtained by Theorem 1 (see also [16] for the case f(u) = u+ the symmetry with respect to *- and +-axis.

Theorem 3. Let be a domain above. Then any least-energy solution to (2) is symmetric with respect to x1 -, x2 -, *-, and +-axis. We do not know uniqueness of solutions to (2), even if we restrict to least-energy solutions. Next, we show some result to support Conjecture 2 in certain domain in which we cannot expect the existence of a least-energy solution. There are several results in this direction (see [15,25,26]). In this paper, we consider the domain , with a parameter , ¿ 0 satisfying the following conditions; (D-1) O ∈ , and , is an unbounded domain which is symmetric with respect to the hyperplane T1 = {x = (x1 ; x ) ∈ R × Rn−1 ; x1 = 0}, (D-2) there exists L ¿ 0 such that     L L   = S, ∩ x = (x1 ; x ); |x1 | 6 ;

, ∩ x = (x1 ; x ); |x1 | 6 2 2 (D-3) there exists d(¿ ,) ¿ 0 such that   d  n−1  ;

, ⊂ (x1 ; x ) ∈ R × R ; |x | ¡ 2 (D-4) , ∩ {x ∈ Rn ; |x| ¡ k} satisIes the uniform cone condition for any suKciently large k ¿ 0. A typical example is the unbounded dumbbell-shaped domain which is included in Sd=2 and consists of the union of two Ad=2 and the thin channel {(x1 ; x ); |x1 | 6 L=2; |x | ¡ ,=2} with , ¡ d (see Fig. 4). For this domain, it is easy to see (e.g. [16,26]) that there is no least-energy solution, namely attains its mountain pass value c( , ) for small ,. Therefore, we must Ind a higher energy solution.

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Fig. 4. An unbounded dumbbell-shaped domain.

Theorem 4. Suppose , satis7es conditions (D-1) – (D-4). Then there exists a su:ciently small ,0 ¿ 0 such that for any , ∈ (0; ,0 ) there exists a solution u = u, to (3) which is symmetric with respect to the hyperplane T1 , and satis7es the following estimates: C1 ,−2=(p−1) 6 u, L∞ ( , ) 6 C2 ,−2=(p−1) ; C3 ,(n−2)=2−2=(p−1) 6 ∇u, L2 ( , ) 6 C4 ,(n−2)=2−2=(p−1) ; where Cj ; j = 1; : : : ; 4 are positive constants independent of ,. The same result has been proved by Byeon [5] (see also [9] for the case n = 2) for bounded dumbbell-shaped domains , . Remark 2. We remark that our method also work even if we relax assumptions (D-2) and (D-3) slightly including V -shaped or H -shaped domains, which are assumed to be symmetric with respect to T1 again, with a thin junction as above. For certain H -shaped domains, we have at least two positive solutions to (3), the one is a least-energy solution and the other is a solution concentrated near the origin in the thin junction.

3. Proofs of Theorems 1–3 In this section, Irst we show Theorem 1 for the case = SR(V ) , since this is the essential part of Theorem 1. Throughout this section, we assume that is a domain with m semi-inInite cylinder as its ends for some m ∈ N. First, we collect some useful known results for the proof of Theorem 1. Under assumptions (f-1) – (f-3), it is known that the energy functional J (u) deIned in Section 2, has a mountain pass structure and we can deIne its mountain value c( ) = inf sup J (0(t)); 0∈1 t∈[0;1]

where 1={0 ∈ C([0; 1]; H01 ( )); 0(0)=0, J (0(1)) ¡ 0}. Moreover, under the additional assumption (f-4), c( ) is characterized as by c( ) =

inf

sup J ((u):

u∈H01 ( );u+ ≡0 (¿0

K. Kurata et al. / Nonlinear Analysis 55 (2003) 83 – 101

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k = ∩ B(O; k)c , where B(O; k)c = {x ∈ Rn ; |x| ¿ k}. For large k ¿ 0, we consider k ), and deIne As before, we can deIne the energy functional J  (u) on H01 ( k ) = inf sup J (0(t)); ck = c( 

0∈1k t∈[0;1]

k

k

where k ) k = {0 ∈ 1; 0(t) ∈ H01 ( 1

for any t ∈ [0; 1]}:

l ) ⊂ H 1 ( k ) for k ¡ l, it is easy to see that ck is no decreasing in k so Since H01 ( 0 that limk → ∞ ck exists. The following is a general criterion due to del Pino and Felmer [10] to assure the existence of a least-energy solution to (2) (see also [16,19]). For similar criterions in other problems, see, e.g., [4,17]. Proposition 1. Assume c( ) ¡ limk → ∞ ck . Then there exists a least-energy solution u, that is J (u) = c( ) and J  (u) = 0. Actually, this proposition is true for any domain for which ∩B(O; k) satisIes the uniform cone property for any large k ¿ 0, although this condition is not mentioned explicitly in [10]. This condition allow us to obtain the uniform constant in Sobolev’s embedding theorem on ∩B(O; k) for any large k ¿ 0 (see [1]). The following lemma is also well known (see, e.g., [24]). Lemma 1. Suppose v ∈ H01 ( ) satis7es J (v) = c( ) and J (v) = supt¿0 J (tv). Then, v must be a least-energy solution to (2). Proof of Theorem 1. Case = SR(V ) : Let = SR(V ) . First, we note that the existence of a least-energy solution uR on SR (see, e.g., [7,10,19]), that is c(SR ) = JSR (uR ) = sup JSR (tuR ): t¿0

By the elliptic regularity theorem, we have uR ∈ C 2 (SR ). It is also known that uR satisIes uR (x1 ; x ) = uR (−x1 ; x ) = uR (x1 ; |x |) for every x = (x1 ; x ) ∈ SR . Now, we may assume that SR(V ) is constructed by cutting a hyperplane which passes the origin and by patching again so that a point A of one part, we say S1 , on the intersection between the plane and SR are transformed into the symmetric point A on the other part, say S2 , with respect to the origin. We may think SR = V1 ∪ V2 and SR(V ) = V1 ∪ V2 , where V1 = V1 and V2 is congruent with V2 and all points x ∈ V2 are transformed to the point x ∈ V2 just by rotating, corresponding to the rotation of the face S2 . Then, there is a natural way to construct a function u˜ on SR(V ) from uR . Namely, deIne u(x ˜  ) = uR (x ) for x ∈ V1 and u(x ˜  ) = uR (x) for x ∈ V2 ,  where x ∈ V2 is the point transformed by x by the rotation above. Then, thanks to the symmetries of uR , we see u˜ is continuously deIned on the face S1 (=S2 ) and hence u˜ ∈ H01 (SR(V ) ). Furthermore, by its construction, we have JS (V ) (t u) ˜ = JSR (tuR ) for every R

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K. Kurata et al. / Nonlinear Analysis 55 (2003) 83 – 101

t ¿ 0 and hence c(SR(V ) ) =

inf

sup JS (V ) (tv)

v∈H01 (SR(V ) ) t¿0

R

˜ 6 sup JS (V ) (t u) t¿0

R

= sup JSR (tuR ) = JSR (uR ) = c(SR ): t¿0

(5)

Claim 1. c(SR(V ) ) ¡ c(SR ) holds. If not, the inequality in (5) should be equal and we have c(SR(V ) ) = sup JS (V ) (t u) ˜ = JS (V ) (u): ˜ t¿0

R

R

˜ The last equality holds because (JS (V ) (t u)=)J SR (tuR ) takes its maximum at t = 1. Thus, R

it follows from Lemma 1 that u˜ should be a least-energy solution to (2) on SR(V ) . By the elliptic regularity theorem we have u˜ ∈ C 2 (SR(V ) ). However, considering the point P near the inner edge of SR(V ) , it is easy see to that u˜ has a jump in its Irst derivative on P because of Hopf’s lemma. This is a contradiction and Claim 1 is proved. Claim 2. c(SR ) 6 limk → ∞ c(SR(V ) ∩ B(O; k)c ) holds.

Since SR(V ) ∩ B(O; k)c is a disjoint union of D1 and D2 , we may think D1 ∪ D2 ⊂ SR and therefore c(SR ) 6 c(SR(V ) ∩ B(O; k)c ). This implies Claim 2. By Claims 1 and 2, we can conclude the existence of a least-energy solution on SR(V ) by using Proposition 1. Claim 1 in the proof above is important, especially in the proof of Theorem 1 for (V ) the case = SR; L. (V ) (V ) Case = SR; L : For = SR; L , we can prove (V ) (V ) lim c(SR; L ) 6 c(SR ):

L→∞

(6)

Once we obtain (6), combining the estimate of Claim 1 above, we obtain for suKciently large L ¿ 0 (V ) (V ) c c(SR; L ) ¡ c(SR ) 6 lim c(SR; L ∩ B(O; k) ); k →∞

in a similar way as the proof above and conclude the existence of a least-energy solution in this case. To show estimate (6), we use a least-energy solution u ∈ H01 (SR(V ) ) to (2) for = SR(V ) . Using a cut-oL function 5L (x) ∈ C ∞ (Rn ) satisfying 0 6 5L (x) 6 1 and (V ) |∇5L (x)| 6 M on Rn for some constant M ¿ 0, 5L (x) = 0 on SR(V ) \SR; L , and 5L (x) = 1 (V ) on SR; L−1 , deIne (V ) uL (x) = u(x)5L (x) ∈ H01 (SR; L ):

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It is easy to see uL → u in H01 (SR(V ) ) as L → ∞. On the other hand, there exists a unique t(uL ) ¿ 0 such that sup JS (V ) (tuL ) = JS (V ) (t(uL )uL ) = JS (V ) (t(uL )uL ); R; L

t¿0

R

R; L

since uL can be seen as uL ∈ H01 (SR(V ) ) by the zero extension. It is known that uL → u in H01 (SR(V ) ) implies t(uL ) → t(u) = 1 as L → ∞ (see, e.g., [19]). It follows that lim sup J

(V ) L → ∞ t¿0 SR; L

(tuL ) = lim JS (V ) (t(uL )uL ) = JS (V ) (u): L→∞

R

R

(V ) Combining c(SR; (V ) (tuL ), we conclude estimate (6). L ) 6 supt¿0 JSR; L General cases: We use the following lemma due to del Pino–Felmer [10].

Lemma 2. Let B be a domain in Rn and let A is a proper subdomain of B. If there exists a least-energy solution to (2) on = A, then we have c(B) ¡ c(A). We prove only in the case that contains SR(V ) , since other cases cane be treated in a similar way. Suppose contains SR(V ) properly, then Lemma 2 implies c( ) ¡ c(SR(V ) ):

(7)

Since we assume that is a domain with m semi-inInite cylinder AR as ends, we can show k ) ¿ c(SR ) c(

(8)

n   for large k. To show this, deIne k; L = k ∩ {x ∈ R ; |x| ¡ L} for large L ¿ 0. Noting  c( k; L ) is decreasing as L → ∞, we claim

  lim c( k; L ) = c( k ):

L→∞

(9)

For any Ixed L ¿ 0, it is easy to see  c(SR ) 6 c( k; L );  because k; L is a disjoint union of Inite cylinder and can be seen as a subset of SR . k ). Combining estimates (7), (8) and c(S (V ) ) ¡ c(SR ) we This implies c(SR ) 6 c( R arrive at the conclusion. Now it remains to show estimate (9). It suKce to show that   if we assume limL → ∞ c( k; L ) ¿ c( k ), then we have a contradiction. Let   6 = lim c( k; L ) − c( k ) ¿ 0: L→∞

k ) and t(uj ) ¿ 0 k ), there exist a sequence {uj }∞ ⊂H 1 ( By the characterization of c( j=1 0 such that k ): J  (t(uj )uj ) = sup J  (tuj ) → c( k

t¿0

k

Let 7L ∈ C0∞ (Rn ) be a function satisfying 0 6 7L (x) 6 1 and |∇7L (x)| 6 M on Rn for some constant M ¿ 0, 7L (x) = 1 on {x ∈ Rn ; |x| 6 L − 1}, and 7L (x) = 0 on

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K. Kurata et al. / Nonlinear Analysis 55 (2003) 83 – 101

k ) and hence t(uj; L ) → t(uj ) {x ∈ Rn ; |x| ¿ L}. Then we have uj; L = 7L uj → uj in H01 ( as L → ∞. We have sup J  (tuj; L ) = J  (t(uj; L )uj; L ) → J  (t(uj )uj ) t¿0

k

k

as L → ∞. This yields  inf c( k; L ) =

k

sup J  (tu)

 u∈H01 ( k;L ) t¿0

k; L

k ) + 6 sup J  (tuj; L ) 6 c( k; L

t¿0

26 3

 ¡ lim c( k; L ): L→∞

This is a contradiction. Proof of Theorem 2. Since we can prove Theorem 2 by using a similar argument as in the proof of Theorem 1 and a scaling argument, we only give a sketch of the proof. Let

be the domain as in Theorem 2. So, especially, is a domain with 2 semi-inInite (V ) cylinder AR as its ends and ⊃ SR; L with a Ixed L ¿ 0. Consider the minimization problem: ∇u2L2 ( ) : e( ) = inf 2 u∈H01 ( );u≡0 uLp+1 ( ) By the transformation v(x) = u(Rx); x ∈ =R ≡ {x ∈ Rn ; x = y=R; y ∈ }, it is easy to see e( ) = Rn−2−(2n)=(p+1) e( =R): Note that  = =R is a domain with two semi-inInite cylinder A1 as its ends and ) . By using a similar argument as in the proof of Theorem 1, one can show

 ⊃ S1;(VL=R

) ) ¡ e(S1 ) for suKciently small R ¿ 0. This implies that e(S1(V ) ) ¡ e(S1 ) and e(S1;(VL=R e( =R) ¡ e(S1 ) and hence e( ) ¡ e(SR ) for suKciently small R ¿ 0. Thus e( ) is attained (see e.g. [16]) and the proof of Theorem 2 is completed.

Proof of Theorem 3. The symmetry with respect to x1 - and x2 -axis can be proved in a similar way as in [5] (see also [6]). It suKces to show the symmetry with respect to *-axis. The symmetry with respect to +-axis can be proved in the same way. Let u be a least-energy solution. Then we know u(x1 ; x2 ) = u(x1 ; −x2 ) = u(−x1 ; x2 ) for every x = (x1 ; x2 ) ∈ . We decompose as a disjoint union of 1 ; 2 and L, where L is the intersection of and *-axis and 1 and 2 are domains which are symmetric with respect to L each other. Now, we consider the mapping T on 2 such that T (x) = x for x ∈ 2 , where x ∈ 2 is the rePection point of x with respect to +-axis. We deIne v on as follows: v(x) = u(x)

for x ∈ 1 ∪ L;

v(x) = u(T (x))

for x ∈ 2 :

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Then we have v ∈ H01 ( ) and J (v) = J (u) = c( ). Moreover, since J (tv) = J (tu) holds for every t ¿ 0, we have sup J (tv) = J (v): t¿0

Then, by Lemma 2 v should be a least-energy solution to (2) on and hence v should be symmetric with respect to x1 - and x2 -axis as before. It follows that u is symmetric with respect to *-axis. 4. Proof of Theorem 4 In this section, we give an outline of the proof of Theorem 4. Although, we basically follow the strategy of Byeon [5], in which the same problem was studied on a bounded domain, we emphasize that how to modify his argument to the problem on unbounded domains. As in [5], by using the scaling v, (x) = ,2=(p−1) u(,x) for x ∈ , = =,, the problem is reduced to Ind a non-trivial positive solution to −5v = vp ;

v(x) ¿ 0

x ∈ , ;

v|@ , = 0;

v(x) → 0(|x| → ∞);

satisfying C1 6 vL∞ ( , ) 6 C2 ;

C3 6 ∇vL2 ( , ) 6 C4

for some positive constants Cj ; j=1; : : : ; 4. Take 6 ¿ 0 so that 86 ¡ L, i.e. 46=, ¡ L=2,. Then, deIne He, = {v ∈ H01 ( , ); v(x1 ; x ) = v(−x1 ; x )} and

  , H = v ∈ He ;

,

p+1

|v|

 d x = 1;

,

e

90 ,|x|

p+1

5, (x)|v(x)|

 dx 6 1 ;

where 5, (x) = 0 for x ∈ {x = (x1 ; x ) ∈ , ; |x1 | 6 26=,} and 5, (x) = ,−3(p+1)=(p−1) for x ∈ {x = (x1 ; x ) ∈ , ; |x1 | ¿ 26=,}. Here 90 ¿ 0 is a positive number to be determined later. Proof of Theorem 4. Step 1: We Irst solve the following minimization problem:  I , = inf |∇v|2 d x: v∈H

,

Lemma 3. There exists a minimizer v = v, (¿ 0) to attain I , for every , ¿ 0. Proof. The proof is done by taking a minimizing sequence and by the standard argument. We note that the compactness can be recovered by the exponential weight, even in an unbounded domain. We omit the details.

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Step 2: We claim that



,

lim sup I 6 I = inf ,→0

2

S1=2

|∇v| d x;





p+1

S1=2

|v|

d x = 1;

v ∈ H01 (S1=2 )

:

The proof is the same as in [5], but we present it for reader’s convenience. It is known that I is attained by V which satisIes V (x1 ; x ) = V (−x1 ; x ) and moreover V and the Irst derivatives of V decays exponentially (see, e.g., [5]). Taking ;, ∈ C0∞ (Rn ) such that 0 6 ;, ; |∇;, | 6 M for some constant M and ;, (x) = 1 on |x| 6 6=, and ;, (x) = 0 on |x| ¿ (6=,) + 2. Then, it is easy to see that for small , we have ;, V ∈ He, and   lim |;, (x)V (x)|p+1 d x = 1: e90 ,|x| 5, (x)|;, (x)V (x)|p+1 d x = 0; ,→0

,

,

Then it is easy to check that  lim |∇(;, V )|2 d x = I; ,→0

,

and thus we have lim sup I , 6 lim



,→0

,→0

,

|∇(;, V )|2 d x = I:

Step 3: We claim that there exists ,0 ¿ 0 such that for every 9 ¿ 0 there exists a constant C ¿ 0 such that  v,p+1 d x ¿ 1 − 9

, ∩{−C¡x1 ¡C}

for every , ∈ (0; ,0 ). The proof of this part is almost the same as in [5] and is done by a concentration-compactness argument. Our modiIcation of the minimization problem I , does not cause any trouble in the argument in [5]. So, we omit the details. We note that the symmetry of the domain plays an important role in this step. Step 4: We claim the following key estimate for the minimizer v, . Lemma 4. There exist positive constants D1 ; D2 and 1 , which are independent of , and 90 such that   √ 56 (10) eD2 ,|x1 | v, (x) 6 D1 e− 1 6=4, x ∈ |x1 | ¿ 2, and √

v, (x) 6 2e−

1 6=4,



26 56 6 |x1 | 6 , 2,

 :

(11)

Proof. We also take the same strategy as in [5], but we need to modify a comparison argument by using a bound state of the Laplacian on unbounded domains. First, we claim that there exist constants *(,) and +(,) such that 5v, + *(,)(v, )p + +(,)exp(90 ,|x|)5, (v, )p = 0:

(12)

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When , exp(90 ,|x|)5, (v, )p+1
d x ¡ 1, it can be concluded by Lagrange’s multiplier theorem as +(,) = 0. When , exp(90 ,|x|)5, (v, )p+1 d x = 1, we note that v, is a minimizer to the minimization problem of two constraints:     2 p+1 p+1 inf |∇v| d x; |v| d x = 1; exp(90 ,|x|)5, |v| dx = 1 :

,

,

,

Thus Lagrange’s multiplier theorem yields the desired result. We claim that +(,) 6 0. This part is the same as in [5], so we omit the proof. Then we have  |∇v, |2 d x = *(,) + +(,);

,

which implies *(,) ¿ 0. The uniform boundedness of *(,) as , → 0 can be proved also as in [5] by using the estimate in step 3. Now, we claim that v, L∞ ( , ) 6 M (n ¿ 3);

v, Lq ( , ∩{|x1 |646=,}) 6 M (n = 2)

(13)

for any q ¿ 2, where M ¿ 0 is a constant independent of , ∈ (0; ,0 ). For the case n ¿ 3, by Sobolev’s embedding theorem and Proposition 3.5 in [5], which is valid even for unbounded domains, we have v, L∞ ( , ) 6 Cv, L2n=(n−2) ( , ) 6 CC  ∇v, L2 ( , ) 6 CC  I 1=2 ; where C and C  are positive constants independent of ,. Here, we used the result in step 2 in the last inequality. For the case n = 2, we take a function 7 ∈ C0∞ (R) such that 0 6 7 6 1; |∇7(t)| 6 1, 7(t) = 1 for |t| 6 46=, and 7(t) = 0 for |t| ¿ (46=,) + 2. Then we can see 7v, ∈ H01 (S1=2 ), since , ∩{|x1 | 6 86=,} ⊂ S1=2 by the deInition of 6. Hence, by Trudinger’s inequality, we have for every q ¿ 2 there exists a constant Cq such that 7v, 2Lq (S1=2 ) 6 Cq 7v, 2H 1 (S1=2 ) 6 C + CCq v, L2 ( , ∩{|x1 |646=,}) + CCq v, L2 ( , ∩{46=,6|x1 |6(46=,)+2}) : On the other hand, by PoincarCe’s inequality on S1=2 , we have 7v, L2 (S1=2 ) 6 C∇(7v, )L2 (S1=2 ) 6 C + Cv, L2 ( , ∩{46=,6|x1 |646=,+2}) 6 C + C|{ , ∩ {46=, 6 |x1 | 6 46=, + 2}| v, L1− p+1 ( , ) 6 C; where 0 ¡  = (p) ¡ 1. Combining these estimates, we obtain that for every q ¿ 2 there exists a constant C which is independent of , ∈ (0; ,0 ) such that 7v, Lq (S1=2 ) 6 C holds. This implies the desired result for the case n = 2. By the elliptic estimate in Theorem 8.25 in [14] (the same estimate also holds if we have a uniform estimate in Lq norm of the potential term with q ¿ n=2, see, e.g., [22]), we have  1=(p+1)  1 , , p+1 (v (y)) dy |v (x)| 6 C |B(x; 1)| B(x;1)

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for every x ∈ , ∩ {|x1 | 6 46=,}, where C is a constant independent of ,. Now we note that, by the estimate in step 3, we may assume that for a Ixed 9 ¿ 0  (v, )p+1 d x 6 9

, ∩{(6=,)−16|x1 |6(36=,)+1}

holds for every , ∈ (0; ,0 ). Now, let =(x) and 1 be the Irst eigenfunction and the Irst eigenvalue of −5 on n − 1-dimensional ball B = {x ∈ Rn−1 ; |x | ¡ 1}. We may assume =(O) = 1 and =(x ) ¿ 0. Note also that =(x ) ¿ 0 if x = (x1 ; x ) ∈ R1; , . Then, by using the estimates above, we may also assume that *(,)(v, (x))p−1 ¡ 3 1 =4

for , ∈ (0; ,0 )

on the region R1; , ≡ , ∩ {6=, 6 |x1 | 6 36=,}: Consider the comparison function √    √  1 1 36 36 x1 + + exp x1 + =(x ): >, (x) = exp − 2 , 2 , Then we have 3 1 >, = 0 5>, + 4 and

in R1; ,

5= + *(,)(v, )p−1 = 6 0

in R1; , :

Thus we obtain 5(>, − v, ) + *(,)(v, )p−1 (>, − v, ) 6 0; >, − v, ¿ 0;

x ∈ R1; , ;

x ∈ @R1; , :

Now, it is easy to see that the function >, (x) − v, (x) z, (x) = =(x ) satisIes =(x )5z, + 2∇= · ∇z, + (5= + *(,)(v, )p−1 =)z, 6 0

in R1; ,

and z, ¿ 0 on @R1; , . Therefore, we can conclude by the maximum principle that v, (x) 6 >, (x); This yields

√

v, (x) 6 2e−

x ∈ R1; , :

1 6=4,

for x ∈ , ∩ {26=, 6 |x1 | 6 56=2,}. Next, we show the estimate on R2; , ≡ , ∩ {|x1 | ¿ 56=2,}:

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Consider the domain (R) = {x ∈ Rn ; |x| ¡ R} ∪ {x = (x1 ; x ) ∈ R × Rn−1 ; |x | ¡ d=2} for large R ¿ d=2. Clearly, , ⊂ (R) . Then it is known (see [11] for related results) that there exists the Irst eigenfunction @R and the Irst eigenvalue 0R such that −5@R = 0R @R ; @R (x) = 0;

@R (x) 6 @R (O);

x ∈ @ (R) ;

x ∈ (R) ;

@R (x) 6 D1 exp(−D2 |x1 |)

for some positive constants D1 and D2 . Take 9 so that 36 ¡ 9 ¡ R. We may take @R so that minB(O; 9) @R (x) = 1. Let ,; (R) = (R) =, and consider  √ 1 6 @R (,x); x ∈ ,; (R) : A, (x) = 2 exp − 4, Then we have 5A, + ,2 0R A, = 0: Noting that minB(O; 9) @R (x) = 1 implies @R (x) ¿ 1 on @R2; , ∩ , , we have A, (x) ¿ v, (x);

x ∈ @R2; , :

On the other hand, we have   (v, )p+1 d x 6

, ∩{|x1 |¿26=,}

, ∩{|x1 |¿26=,}

e90 ,|x| (v, )p+1 d x 6 ,3(p+1)=(p−1) :

By applying Theorem 8.25 in [14] again for x ∈ R2; , and noting B(x; 1) ∩ , ⊂ , ∩ {|x1 | ¿ 26=,}, we obtain v, (x) 6 C,3=(p−1) : Here, in the case n = 2, we use the boundedness of (v, )p−1 L(p+1)=(p−1) ( , ) to control the uniform boundedness of the constant appeared in the generalized version of Theorem 8.25 of [14] (see [22]). Now, we obtain 5(A, − v, ) + *(,)(v, )p−1 (A, − v, ) 6 0; A, − v, ¿ 0;

x ∈ R2; , ;

x ∈ @R2; ,

,

and A, (x) − v (x) → 0 as |x| → ∞. Let =˜ and ˜1 be the Irst eigenfunction and the Irst eigenvalue of the Laplacian on {x ∈ Rn−1 ; |x | ¡ d}. Then we have ˜  ) 6 5=(,x ˜  ) + ,2 ˜1 =(,x ˜ ) = 0 5=(,x ˜  ) + *(,)(v, )p−1 =(,x ˜  ) ¿ ( ¿ 0 with a positive constant ( which is independent of ,. Let and inf x∈R2; , =(,x A, (x) − v, (x) z , (x) = : =(,x ˜ ) Then z , (x) → 0 as |x| → ∞ and we can see that the maximum principle can be applied to z , on R2; , as before to obtain z , ¿ 0 in R2; , . This yields the desired estimate on R2; , . By estimates (10), (11), (13) and Proposition 3.5 in [5], now we have the uniform boundedness of v, on , even in the case n = 2.

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Step 5: First, note that there exists a constant D3 which is independent of , that |x| 6 D3 |x1 | on , ∩ {|x1 | ¿ 26=,}. Take 90 ¿ 0 so that D3 90 ¡ D2 (p + 1), where D2 is the constant appeared in the estimate of step 4. Then, dividing , into two parts and using estimates in step 4, we can easily see  √ e90 ,|x| 5, (x)(v, (x))p+1 d x 6 C,−(n+(3(p+1))=(p−1)) e−((p+1) 1 6)=4, → 0

,

as , → 0. Thus there exists a constant ,0 ¿ 0 such that  e90 ,|x| 5, (x)(v, (x))p+1 d x ¡ 1

,

holds for , ∈ (0; ,0 ). Therefore, deIning u, (x) = (I , )1=(p−1) v, (x), we obtain −5u, = (u, )p : Note that we can see from the estimate in Step 4 lim inf , → 0 I , ¿ I . Then the uniform lower bounds for u, follow from estimates in Steps 2– 4. Acknowledgements The Irst author was supported in part by grant-in-aid for ScientiIc Research (C), The Ministry of Education, Science, Sports and Culture, Japan, No. 13640183. References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] C.J. Amick, J.F. Toland, Nonlinear elliptic eigenvalue problems on an inInite strip—Global theory of bifurcation and asymptotic bifurcation, Math. Ann. 262 (1983) 3131–3342. [3] A. Bahri, P.L. Lions, On the existence of a positive solution of semi-linear equations in unbounded domains, Ann. Inst. H. PoincarCe Anal. Non LinCeaire 14 (1997) 365–413. [4] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [5] J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial DiLerential Equations 22 (1997) 1731–1769. [6] K.J. Chen, K.C. Chen, H.C. Wang, Symmetry of positive solutions of semi-linear elliptic equations in inInite strip domains, J. DiLerential Equations 148 (1998) 1–8. [7] C.N. Chen, S.Y. Tzeng, Some properties of Palais–Smale sequences with applications to elliptic boundary value problems, Electron. J. DiLerential Equations (17) (1999) 1–29. [8] A. Dall’aglio, M. Grossi, A perturbation theorem for the equation −5u+ u=up in unbounded domains, Nonlinear Anal. 28 (1997) 1867–1877. [9] E.N. Dancer, On the inPuence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann. 285 (1989) 647–669. [10] M.A. del Pino, P.L. Felmer, Least energy solutions for elliptic equations in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 195–208. [11] P. Duclos, P. Exner, Curvature-induced bound states in quantum wave guides in two and three dimensions, Rev. Math. Phys. 7 (1995) 73–102. [12] M.J. Esteban, Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex ring, Nonlinear Anal. 7 (1983) 365–379. [13] M.J. Esteban, P.L. Lions, Existence and non-existence results for semi-linear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh A 93 (1982) 1–12.

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