EXISTENCE OF POSITIVE SOLUTIONS FOR A SEMILINEAR

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 9, September 2006, Pages 2585–2592 S 0002-9939(06)08405-X Article electronically published on February 17, 2006

EXISTENCE OF POSITIVE SOLUTIONS FOR A SEMILINEAR ELLIPTIC PROBLEM WITH CRITICAL SOBOLEV AND HARDY TERMS NORIMICHI HIRANO AND NAOKI SHIOJI (Communicated by David S. Tartakoff) Abstract. Let N ≥ 4, let 2∗ = 2N/(N − 2) and let Ω ⊂ RN be a bounded domain with a smooth boundary ∂Ω. Our purpose in this paper is to consider the existence of solutions of the problem: ⎧ 2∗ −1 u ⎪ in Ω, ⎨ −∆u − µ |x|2 = |u| ⎪ ⎩

u u

> =

0 0

in Ω, on ∂Ω,

where 0 < µ < ( N 2−2 )2 .

1. Introduction Let N ≥ 4, let 2∗ = 2N/(N − 2) and let Ω ⊂ RN be a bounded domain with a smooth boundary ∂Ω and 0 ∈ Ω. In the present paper, we consider the existence of solutions of the problem ⎧ ∗ u ⎪ = |u|2 −1 in Ω, ⎨ −∆u − µ |x| 2 (Pµ ) u > 0 in Ω, ⎪ ⎩ u = 0 on ∂Ω, where µ is a given positive constant. These kind of equations give nonlinear Schr¨ odinger equations with the field having singularity at its origin. The existence of solutions of semilinear elliptic boundary value problems with Hardy terms has been studied by many authors (cf. [4], [5], [6], [8], [10] and [12]). It is known that problem (Pµ ) has no nontrivial solution when Ω is star shaped (cf. [1]) for any µ ≥ 0. On the other hand, in the case that Ω has nontrivial topology, problem (P0 ) has been investigated by several authors. In [9], Kazdon and Warner proved the existence of a nontrivial solution of (P0 ) in the case that Ω is an annulus. In [2], Bahri and Coron established the existence of a nontrivial solution of (P0 ) in the case that Ω has nontrivial topology. Our purpose in the present paper is to show the existence of a solution of (Pµ ) with µ > 0 when domain Ω has nontrivial topology. We now state our main result. Received by the editors August 31, 2004 and, in revised form, March 21, 2005. 2000 Mathematics Subject Classification. Primary 35J65, 35J20. Key words and phrases. Critical Sobolev, Hardy inequality, semilinear elliptic problem. This work was partially supported by the Heisei16 joint research project fund in the Graduate School of Environment and Information Sciences of Yokohama National University. c 2006 American Mathematical Society

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NORIMICHI HIRANO AND NAOKI SHIOJI

Theorem 1.1. Suppose that Ω is not contractible. Then there exists µ0 ∈ (0, ∞) such that for each µ ∈ (0, µ0 ), there exists a solution of (Pµ ). 2. Preliminaries 2 p p Let H = H01 (Ω). We put v = Ω |∇v| dx for v ∈ H and |v|p = Ω |v| dx for p ≥ 2 and v ∈ Lp (Ω). For u, v ∈ H, we put u, v = Ω uv dx. We use the same p symbols · , ||p , and ·, · in the case Ω = RN . For each A ⊂ RN and x ∈ RN , d(x, A) denotes the distance of x from A. We denote by Br (x) the open ball in RN centered at x with radius r. For subsets A, B ⊂ RN , A ∼ = B implies that A i and B are homotopy equivalent. For each d > 0, Ω and Ω d d stand for subsets N N i of R defined by Ωd = x ∈ R : d(x, Ω) < d and Ωd = {x ∈ Ω : d(x, ∂Ω) > d} , respectively. For each a ∈ R and a functional F : H → R, we denote by Fa the level set Fa = {v ∈ H : Fa (v) ≤ a} . The Hardy inequality states that 2

N −2 u2 2 ≤ | u| 2 2 N N R |x| R ∗ holds for u ∈ D1,2 (RN ) = v ∈ L2 (RN ) : |∇v| ∈ L2 (RN ) (cf. [12]). For each bounded domain U ⊂ RN and µ ≥ 0, we define a functional I (U,µ) on H01 (U ) by

1 µ |u|2 1 2∗ 2 ( |∇u| − − ∗ u+ )dx for u ∈ H01 (U ). I (U,µ) (u) = 2 2 |x| 2 U 2 2

N

I (R ,µ) (v) is defined by the same way as above with U = RN and v ∈ D1,2 (RN ). Here u+ (x) = max {u(x), 0} for x ∈ U. Then the solutions of (Pµ ) correspond to critical points of the functional I (Ω,µ) . Throughout the rest of this paper, we assume √ that 0 ≤ µ < µ = ( N2−2 )2 . Denote β = µ − µ. For each (z, ε) ∈ RN × R+ , we put (µ) u(z,ε) (x)

=

Cε |x − z|

N −2 2 −β

N −2 4 4β

(ε + |x − z| N −2 )

N −2 2

for x ∈ RN ,

where C is a suitable positive constant. Terracini [12] showed that for each ε > 0, (µ) (µ) u(0,ε) is a solution of problem (Pµ ) with Ω = RN . That is, for each ε > 0, u(0,ε) is a critical point of I (R

N

,µ)

with the same critical value. We put cµ = I (R

N

,µ)

(µ)

(u(0,ε) ) (0)

for each µ ≥ 0 and ε > 0. It is also known that for each z ∈ RN and ε > 0, u(z,ε) is N

a critical point of I (R ,0) with the same critical value c0 . One can see that cµ < c0 for each µ > 0. We put

2 µ |v| 2 2∗ 1,2 N (| v| − )dx = |v| dx Sµ = v ∈ D (R ) {0} : |x|2 RN RN for each µ ≥ 0. We put Sµ (Ω) = Sµ ∩ H. One can see that for each µ ≥ 0 and v ∈ D1,2 (RN ) {0} , there exists tv,µ > 0 such that tv,µ v ∈ Sµ (cf. [7]). From the definition of Sµ , we find that for all v ∈ Sµ . (2.1) ∇I (Ω,µ) (v), v = 0


ELLIPTIC PROBLEM WITH CRITICAL SOBOLEV AND HARDY TERMS

2587

Then each nontrivial critical point of I (Ω,µ) is contained in Sµ . For each µ, we have

1 µ |v|2 1 2 2∗ cµ = inf (| v| − |v| dx : v ∈ Sµ . 2 )dx − 2∗ 2 RN |x| RN From the definition of Sµ , we also find

µ |v|2 2 (| v| − (2.2) cµ = inf m0 2 ) : v ∈ Sµ , |x| RN ∗

where m0 = 22·2−2 ∗ . It also follows from the Hardy inequality that cµ −→ c0 as µ −→ 0. Then we can choose µ ∈ (0, µ) such that 2cµ > c0 for each µ ∈ (0, µ ). Lemma 2.1. Let µ ∈ (0, µ ). If q ∈ (0, 1) and v ∈ D1,2 (RN ) such that

|v|2 cµ 2 (| v| − µ 2 )dx ≤ q , m N 0 |x| R then

RN

(| v|2 − µ

|v|

2

|x|2

∗

∗

− |v|2 ) ≥ (1 − q −(2−2

)/2

) RN

(| v|2 − µ

|v|

2

|x|2

).

Proof. Let µ ∈ (0, µ ) and q ∈ (0, 1). Let v ∈ D1,2 (RN ) {0} such that

2 |v| cµ (| v|2 − µ 2 ) ≤ q . m N 0 |x| R We put t = tv,µ . Then by the definition,

2 ∗ ∗ |v| cµ 2 t (| v|2 − µ 2 ) = t2 |v|22∗ ≥ m 0 |x| RN and then t ≥ q −1/2 . Therefore

2 2 ∗ ∗ |v| |v| (| v|2 − µ 2 − |v|2 ) = (1 − t2−2 ) (| v|2 − µ 2 ) |x| |x| RN RN

∗ |v|2 2 (| v| − µ 2 ). ≥ (1 − q −(2−2 )/2 ) |x| RN

Lemma 2.2. Let µ ∈ (0, µ ) and assume that there exists no solution of (Pµ ). Let {vn } ⊂ Sµ (Ω) satisfy (2.3)

lim I (Ω,µ) (vn ) = c < c0 and

n→∞

lim I (Ω,µ) (vn ) = 0.

n−→∞

+ N Then there exist {ε n } ⊂ R and {x n } ⊂ R such that limn−→∞ |xn | = 0, limn−→∞ εn (µ) = 0 and limn→∞ vn − u(xn ,εn ) = 0.

Proof. The proof of the assertion is based on the standard argument for the case that µ = 0 (cf. Struwe [11]). Fix µ ∈ (0, µ ) and assume that there exists no solution of (Pµ ). Let {vn } ⊂ Sµ (Ω) be a sequence satisfying (2.3). From the definition of Sµ ,

2 |vn | 2 for n ≥ 1. I (Ω,µ) (vn ) = m0 (| vn | − µ 2 )dx |x| Ω


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Then since limn→∞ I (Ω,µ) (vn ) = c, we have by the Hardy inequality that {vn } is bounded in H. Then we may assume that vn −→ v weakly in H, vn −→ v strongly ∗ in L2 −1 (Ω) and vn −→ v a.e. on Ω. Then we find that v is a solution of problem (Pµ ). If v = 0, this contradicts the assumption. Therefore we have that vn −→ 0 ∗ strongly in L2 −1 (Ω) and vn −→ 0 a.e. on Ω. Let k0 , k1 be positive integers such that B2 (0) can be covered by k0 balls with radius 1, Ω can be covered by k1 balls with radius 1, and c0 < 2k0 < k1 . Let {(xn , rn )} ⊂ RN × R+ be a sequence such that

2 |vn | c0 2 N (| vn | − µ 2 ) = for some x ∈ R rn = min r > 0 : m0 k1 |x| Br (x) and

(| vn |2 − µ

m0 Brn (xn )

|vn |2 2

|x|

)=

c0 k1

for each n ≥ 1. Since vn −→ 0 a.e. on Ω, one can see that rn −→ 0 as n −→ ∞. Here we put wn (x) = rn(N −2)/2 vn (xn + rn x) for x ∈ RN . Then wn ∈ H01 (Ωn ), where Ωn = x ∈ RN : xn + rn x ∈ Ω . It also follows from (2.3) that rn wn 2∗ −1 −→ 0 as n −→ ∞. −∆wn − µ 2 − wn |xn + rn x| We also have that

|vn |2 r 2 |wn |2 c0 (| vn |2 − µ 2 ) = (| wn |2 − µ n 2) = k m 1 0 |x| |x + r x| Brn (xn ) B1 (0) n n for each n ≥ 1. Then since rn ≤ 1 for each n ≥ 1, wn −→ w0 ∈ D1,2 (RN ) weakly in D1,2 (RN ). We will see that wn −→ w0 strongly in D1,2 (Ω ) for any bounded domain Ω ⊂ RN . It is sufficient to show the case that Ω = B1 (0). For each n ≥ 1, we can construct functions {ϕn } such that ϕn = wn − w0 on Bρ (0) for each n ≥ 1, where ρ ∈ (1, 2) and ϕn D1,2 (RN Bρ (0)) −→ 0 as n −→ ∞ (cf. the proof of Lemma 3.3 of [11]). Then by using the Brezis-Lieb lemma we have that

2

lim

n−→∞

RN

(| ϕn | −µ

rn2 |ϕn |

2

|xn + rn x|2

= lim n−→∞

) 2

(| ϕn | − µ

Bρ (0)

2

(| wn | − µ

= lim

n−→∞

Bρ (0)

≤

rn2 |wn |2

k0 c0 . k1 m0


2)

|xn + rn x| 2

B2rn (xn )

2)

|xn + rn x|

(| vn |2 − µ

≤ lim

n−→∞

rn2 |ϕn |2

|vn |

2

|x|

)


Since

k0 c0 k 1 m0

0 such that rn2 wn ϕn

( wn ϕn − µ

n−→∞

|xn + rn x|

(| (wn − w0 )|2 − µ rn2

2

(| ϕn | − µ

n−→∞

Bρ (0)

− wn2

−1

rn2 |wn − w0 |2

∗

|xn + rn x|

|ϕn |2

ϕn )

− |wn − w0 |2 )

2

Bρ (0)

= lim

∗

2

= lim

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2∗

2

|xn + rn x|

− |ϕn | )

|ϕn |2 2 2∗ (| ϕn | − µ 2 − |ϕn | ) n−→∞ RN xn r n + x

|ϕn |2 ≥ lim C (| ϕn |2 − µ 2 ). n−→∞ xn RN r n + x

= lim

Then we have by the Hardy inequality that limn−→∞ | ϕn |22 = 0 and then wn −→ w0 strongly in D1,2 (B1 (0)). Therefore we have that wn −→ w strongly in H01 (Ω ) for each bounded domain Ω ⊂ RN . We may assume by subtracting subsequences that xn /rn −→ x0 ∈ RN or |xn /rn | −→ ∞ holds as n −→ ∞. If |xn /rn | −→ ∞ holds, then recalling that wn 2∗ −1 (2.4) −∆wn − µ −→ 0 as n −→ ∞, 2 − wn xn r n + x we have w0 ∈ D1,2 (RN ) is a solution of the problem ∗

−∆w = w2

−1

on RN .

Then we find c0 > lim I (Ω,µ) (vn ) ≥ lim I (R n→∞

n−→∞

N

,0)

(w0 ) = c0 .

This is a contradiction. Therefore we have that xn /rn −→ x0 ∈ RN . This implies that limn−→∞ |xn | = 0, and w0 is a positive solution of problem −∆w − µ

w

∗

2

|x0 + x|

= w2

−1

on RN .

By the translation, we may assume without any loss of generality that x0 = 0. Then N N 2 limn−→∞ I (R ,µ) (wn ) ≥ I (R ,µ) (w0 ) = cµ . Suppose that limn−→∞ | (wn − w0 )|2 > 0. Then noting that 2

2

2

lim | (wn − w0 )|2 + | w0 |2 = lim | wn |2 ,

n−→∞

n−→∞

∗

∗

∗

lim |wn − w0 |22∗ + |w0 |22∗ = lim |wn |22∗ ,

n−→∞

n−→∞

and

|wn − w0 |2

lim

n−→∞

RN

2

|x|

|w0 |2

+ RN

|x|

2

= lim

n−→∞


Ω

|wn |2 2

|x|

,

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we find

2

lim

n−→∞

RN

(| (wn − w0 )|2 − µ

|wn − w0 | 2

|x|

∗

− |wn − w0 |2 ) = 0.

Then we have lim inf I (R

N

,µ)

n−→∞

(wn − w0 ) ≥ cµ .

Therefore we find that lim inf I (R

N

n−→∞

,µ)

(wn ) ≥ I (R

N

,µ)

(w0 ) + lim inf I (R

N

,µ)

n−→∞

(wn − w0 ) ≥ 2cµ .

This is a contradiction. Thus we have wn −→ w0 strongly in D1,2 (RN ) and N (µ) I (R ,µ) (w0 ) = cµ . Therefore w0 = u(0,ε) for some ε > 0. Then from the defini tion of wn , we find that the assertion holds. In case µ = 0, it is known that analogous results to Lemma 2.2 hold. We state a lemma for the case µ = 0. The proof is quite same as that of Lemma 2.2 above. Lemma 2.3. Let {vn } ⊂ H be a sequence such that 2∗

2

lim | vn | = lim |vn |2∗

n−→∞

n−→∞

and

lim I (Ω,0) (vn ) ≤ c0 .

n−→∞

Then there {(zn , εn )} ⊂ RN × R+ such that limn−→∞ εn = 0 and exists a sequence (0) limn−→∞ vn − u(zn ,εn ) = 0. Here we choose a positive number d > 0 such that Ωd ∼ = Ω. We put 2 x |∇v(x)| dx for each v ∈ H \ {0} . β(v) = Ω 2 |∇v(x)| dx Ω Lemma 2.4. There exists µ0 ∈ (0, µ ) such that for each µ ∈ (0, µ0 ), and each v ∈ Sµ (Ω) with I (Ω,µ) (v) < c0 , β(v) ∈ Ωd . Proof. Suppose to the contrary that there exists a sequence {(vn , µn )} ⊂ H × R+ such that limn−→∞ µn = 0, vn ∈ Sµn , I (Ω,µn ) (vn ) < c0 and β(vn ) ∈ / Ωd for all n ≥ 1. We may assume without any loss of generality that β(vn ) −→ z0 ∈ RN Ωd . Since µn −→ 0, we have by Hardy’s inequality that

vn2 as n −→ ∞. µn 2 −→ 0 Ω |x| Then we find that 2∗

2

lim | vn | = lim |vn |2∗

n−→∞

n−→∞

and

lim I (Ω,0) (vn ) ≤ c0 .

n−→∞

N Then by Lemma 2.3, we find that there exists a sequence {(zn , εn )} ⊂ R × (0) R+ such that limn−→∞ εn = 0 and limn−→∞ vn − u(zn ,εn ) = 0. From the assumption, we have that / Ωd , we have that limn−→∞ zn = z0 . But since z0 ∈ limn−→∞ vn − u(zn ,εn ) > 0. This is a contradiction.

Here we choose d1 > 0 such that Ω ∼ = Ωid1 . We put 2 λ = inf µ/ |x| : x ∈ Ωd > 0.



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We fix a function ϕ ∈ C ∞ (R+ : [0, 1]) such that ϕ(x) = 1 for x ∈ [0, d1 /2] and ϕ(x) = 0 for x ∈ [d1 , ∞). For each (z, ε) ∈ RN × R+ , we define a function v(z,ε) by (0)

v(z,ε) (x) = τ(z,ε) ϕ(x − z)u(z,ε) (x)

for x ∈ RN ,

where τ(z,ε) is a positive constant such that ∗ v(x,ε) 2 − λ v(x,ε) 2 = v(x,ε) 2∗ . 2 2 Lemma 2.5. Let µ ∈ (0, µ0 ). Then there exists ε > 0 such that sup I (Ω,µ) (tv(z,ε) ,µ v(z,ε) ) : z ∈ Ωid1 < c0 . Proof. We put

1 1 2 2 2∗ (| v| − λ |v| ) − ∗ |v| 2 Ω 2 Ω for v ∈ H. Then by the results of Brezis and Nirenberg [3], we have that c0 − λCε + o(ε) for N ≥ 5, (2.5) Q(v(z,ε) ) = for N = 4. c0 − λCε log ε + O(ε) Q(v) =

Then we can choose ε > 0 sufficiently small so that Q(v(z,ε) ) < c0 for all z ∈ Ωid1 . 2 Let z ∈ RN . We put t = tv(z,ε) ,µ Then noting that µ/ |x| > λ for x ∈ Ω, we have that

2 v(z,ε) 2 2 2 2 t t2 ( v(z,ε) − λ v(z,ε) ) > ( v(z,ε) − µ 2 ) 2 Ω 2 Ω |x| ∗

∗ t2 v(z,ε) 2 = 2 Ω ∗

2 2 t2 = ( v(z,ε) − λ v(z,ε) ). 2 Ω Then t < 1. Therefore we have that I

(Ω,µ)

2

2 v(z,ε) t2 (2∗ − 2) (tv(z,ε) ) = ( v(z,ε) 2 − µ 2 ) 2 · 2∗ |x| Ω 2 2 (2∗ − 2) ≤ ( v(z,ε) 2 − λ v(z,ε) 2 ) ∗ 2·2 = Q(v(z,ε) ) < c0 ,

and this completes the proof.

Proof of Theorem 1.1. Fix µ ∈ (0, µ0 ). Let ρ : Sµ (Ω) × [0, ∞) −→ Sµ (Ω) be a pseudo-gradient flow associated with I (Ω,µ) (cf. [11]). That is, ρ satisfies: (1) For s, t ∈ R+ with t > s and v ∈ Sµ with I (Ω,µ) (v) = 0, I (Ω,µ) (ρ(t, v)) < I (Ω,µ) (ρ(s, v)); lim I (Ω,µ) (ρ(t, v)) > −∞ implies that lim I (Ω,µ) ((t, v)) = 0. t−→∞ Here we put V = tv(z,ε) ,µ v(z,ε) : z ∈ Ωid1 , where ε is the positive constant obtained in Lemma 2.5. From the definition of v(z,τ ) , we have that v(z,ε) ∈ H01 (Ω) for z ∈ Ωid1 . Then V ⊂ Sµ (Ω). Then since I (Ω,µ) (ρ(t, v)) : t ≥ 0 is bounded from below for each v ∈ V , we have that limn−→∞ I (Ω,µ) (ρ(t, v)) = 0 for each v ∈ V. (2)

t−→∞


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Then we have by Lemma 2.2 that for each v ∈ V, there exists {(zt , εt )}t≥0 ⊂ Ω×R+ (µ) such that limt−→∞ |zt | = 0 and limt−→∞ ρ(t, v) − u(zt ,εt ) = 0. This implies that lim β(ρ(t, v)) = 0 ∈ Ω

t−→∞

for all v ∈ V.

We also have by Lemma 2.4 that {β(ρ(t, v)) : v ∈ V } ⊂ Ωd . Since {β(ρ(0, v)) : v ∈ V } = Ωid1 , we have that Ωid1 is contractible in Ωd . This contradicts the assumption that Ωid1 ∼ = Ω ∼ = Ωd and that Ω is not contractible. Then we obtain that there exists a positive solution u of (Pµ ) in Sµ . References 1. J.P.G. Azorero and I.P. Alonso, Hardy inequalities and some critical elliptic and parabolic problmes, J. Differential. Equations 144 (1998), 441–476. MR1616905 (99f:35099) 2. A. Bahri and M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294. MR0929280 (89c:35053) 3. H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exonents, Comm. Pure Appl. Math. 36 (1983), 437–477. MR0709644 (84h:35059) 4. P. Caldiroli and A. Malchiodi, Singular elliptic problems with critical growth, Comm. Partial Diff. Equations 27 (2002), 847–876. MR1916550 (2003f:35094) 5. A. Ferrero and F. Gazalla, Existence of solutions for singular critical growth semilinear elliptic equations, J. Diff. Equations 177 (2001), 494–522. MR1876652 (2002m:35068) 6. N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. AMS 352 (2000), 5703–5743. MR1695021 (2001b:35109) 7. N. Hirano, Multiple existence of solutions for semilinear elliptic problems on a domain with a rich topology, Nonlinear Analysis TMA 29 (1997), 725–736. MR1455061 (98d:35056) 8. E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Equations 156 (1999), 407–426. MR1705383 (2000f:35053) 9. J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567–597. MR0477445 (57:16972) 10. F. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Diff. Equations 190 (2003), 524–538. MR1970040 (2004c:35138) 11. M. Struwe, Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems, Springer, 1996. MR1411681 (98f:49002) 12. S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Equations 1 (1996), 241–264. MR1364003 (97b:35057) Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan E-mail address: [email protected] Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan
