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3 Sublinear Lane–Emden Systems with Singular Data

3 Sublinear Lane–Emden Systems with Singular Data

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156



4 Singular Lane–Emden–Fowler Equations and Systems



(ii) If 0 < a < 1 then there exists c > 0 such that u ≤ cδ (x) in Ω .

(iii) If a = 1 and τ > 1 then, there exist c > 0 and A > diam(Ω ) such that

u ≤ cδ (x) logτ



A

δ (x)



in Ω .



In particular, for any 0 < ε < 1 there exist c > 0 such that u ≤ cδ (x)1−ε in Ω .

(iv) If 1 < a < 2 then, there exists c > 0 such that u ≤ cδ (x)2−a in Ω .

We shall prove only (i), the proof of (ii) is similar. First, we fix τ > 0 such that

(p + τ )q < 1. We divide our argument into three cases according to the boundary behavior of the solutions to some singular elliptic inequalities as described in Lemma

4.22.

Case 1: 1 < a − p < 2. By Lemma 4.22, there exists C > 0 such that:

• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−a+p , w > 0



in Ω ,



(4.65)



on ∂ Ω ,



w=0

satisfies

w(x) ≤ Cδ (x)2−a+p



in Ω .



(4.66)



• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−b+q(2−a+p) , w > 0



in Ω ,



(4.67)



on ∂ Ω ,



w=0

satisfies

w(x) ≤ Cδ (x)



in Ω .



(4.68)



C < min{M τ , M 1−(p+τ )q }



(4.69)



We fix M > 1 with the property



and define

A :=



(u, v) ∈ C(Ω ) × C(Ω ) :



0 ≤ u ≤ M p+τ δ (x)2−a+p



in Ω



0 ≤ v ≤ M δ (x)



in Ω



.



4.3 Sublinear Lane–Emden Systems with Singular Data



157



For any (u, v) ∈ A , we consider (Tu, T v) the unique solution of



−Δ (Tu) = δ (x)−a v p , Tu > 0 in Ω ,







−Δ (T v) = δ (x)−b uq , T v > 0 in Ω ,







Tu = T v = 0

on ∂ Ω ,



(4.70)



and define F : A → C(Ω ) × C(Ω ) by

F (u, v) = (Tu, T v)



for any (u, v) ∈ A .



(4.71)



Remark that (4.70) has a unique solution, in other words, F is well defined. Indeed

for the existence of Tu we remark that 0 ≤ δ (x)−a v p ≤ cδ (x)−a+p in Ω . Therefore,

by Lemma 4.22(iv) we find that w ≡ 0 and w = Aϕ12−a , A > 1 large, are sub and

supersolutions respectively. Therefore, there exists Tu ∈ C2 (Ω ) ∩ C(Ω ) such that

−Δ (Tu) = δ (x)−a v p in Ω and Tu = 0 on ∂ Ω . The uniqueness of Tu and the fact

that Tu > 0 in Ω follows from the standard maximum principle. The existence and

uniqueness of T v is similar.

As in the previous section we next prove that F is compact and continuous and

that F (A ) ⊆ A . By the Schauder fixed point theorem we then obtain that F has a

fixed point which is a solution of (4.64).

Case 2: a − p = 1. We fix ε > 0 small enough such that b − q(1 − ε ) < 1. By Lemma

4.22(ii)–(iii), there exists C > 0 such that

• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−1 , w > 0

w=0



in Ω ,

on ∂ Ω ,



satisfies

w(x) ≤ Cδ (x)1−ε



in Ω .



• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−b+q(1−ε ) , w > 0



on ∂ Ω ,



w=0

satisfies

w(x) ≤ Cδ (x)



in Ω ,



in Ω .



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4 Singular Lane–Emden–Fowler Equations and Systems



We next proceed as before by considering

A =



(u, v) ∈ C(Ω ) × C(Ω ) :



0 ≤ u(x) ≤ M p+τ δ (x)1−ε



in Ω



0 ≤ v(x) ≤ M δ (x)



in Ω



,



where M > 1 is a constant that fulfills (4.69).

Case 3: a − p < 1. In this case −b + q < 1 and by Lemma 4.22 there exists C > 0

such that

• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−a+p , w > 0



in Ω ,

on ∂ Ω ,



w=0

satisfies

w(x) ≤ Cδ (x)



in Ω .



• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≤ δ (x)−b+q(1−ε ) , w > 0



in Ω ,

on ∂ Ω ,



w=0

satisfies

w(x) ≤ Cδ (x)



in Ω .



We next proceed as in Case 1 by considering

A =



(u, v) ∈ C(Ω ) × C(Ω ) :



0 ≤ u(x) ≤ M p+τ δ (x)



in Ω



0 ≤ v(x) ≤ M δ (x)



in Ω



.



This finishes the proof of Theorem 4.21.



4.3.2 Case p > 0 and q < 0

In this section we shall be concerned with the case p > 0 > q. First, we obtain the

following nonexistence result.

Theorem 4.23 Assume p > 0 > q and one of the following holds:

(i) a − p ≥ 2 or b ≥ 2.

(ii) 0 < a < 1 and b − q ≥ 2.



4.3 Sublinear Lane–Emden Systems with Singular Data



159



(iii) a = 1 and b > 1 and b ≥ 2 + q.

(iv) 1 < a < 2 and b − q(2 − a) ≥ 2.

Then, the system (4.64) has no solutions.

Proof. (i) Assume a − p ≥ 2. By Lemma 4.7 we have v ≥ cδ (x) in Ω , for some

c > 0. Using this estimate in the first equation of (4.64) we find −Δ u ≥ c1 δ (x)a−p

in Ω , for some c1 > 0, which is impossible by Corollary 4.9.

If b ≥ 2, we use the second equation of (4.64) to derive

−Δ v ≥ u



q

−b

∞ δ (x)



in Ω ,



which is impossible according to Corollary 4.9.

(ii) From the first equation of (4.64) and p > 0 we deduce

−Δ u ≤ v



p

−a

∞ δ (x)



in Ω .



Since 0 < a < 1, Lemma 4.22(ii) yields u ≤ c0 δ (x) in Ω , for some c0 > 0. Using

this estimate in the second equation of (4.64) we deduce −Δ v ≥ c1 δ (x)−b+q in Ω .

Since b − q ≥ 2, we arrive at a contradiction according to Corollary 4.9.

(iii) Let τ > 1. Since −Δ u ≤ v



p

−1

∞ δ (x)



u ≤ cδ (x) logτ



in Ω , there exists c > 0 such that



A

δ (x)



in Ω .



Thus,

−Δ v = δ (x)−b uq ≥ Cδ (x)−b+q logτ q



A

δ (x)



in Ω ,



where C is a positive constant. By Theorem 4.8 we now deduce

1



t −b+q+1 logτ q



0



A

dt < ∞

t



which, in view of the fact that b ≥ q + 2 and b > 1, yields b − q = 2 and τ q <

−1. Since τ > 1 was arbitrary, it follows that q ≤ −1 so b = q + 2 ≤ 1, which is

impossible.

(iv) As before, from the first equation of (4.64) we have

−Δ u ≤ v



p

−a

∞ δ (x)



in Ω .



From Lemma 4.22(iv) we find u ≤ c2 δ (x)2−a in Ω , where c2 is a positive

constant. Using this last inequality in the second equation of (4.64) we derive



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4 Singular Lane–Emden–Fowler Equations and Systems



−Δ v ≥ c3 δ (x)−b+q(2−a) in Ω , which is impossible by Corollary 4.9. This ends the

proof of Theorem 4.23.

Theorem 4.24 Let p > 0 > q satisfy pq > −1.

(i) If b − q < 1 then the system (4.64) has solutions if and only if a − p < 2.

(ii) If 0 < a < 1 then the system (4.64) has solutions if and only if b − q < 2.

Furthermore, in both the above cases the system (4.64) has a unique solution.

Proof. (i) If a − p ≥ 2, then by Theorem 4.23 there are no solutions of system

(4.64). Suppose next that a − p < 2. The proof is similar to that for Theorem 4.21.

Fix τ > 0 such that (p + τ )q > −1 and assume first that 1 < a − p < 2. According

to Proposition 4.10(i) and Lemma 4.22(iv) there exist c1 > c2 > 0 such that

• Any function w ∈ C2 (Ω ) ∩C(Ω ) that satisfies (4.65) also fulfills

w(x) ≤ c1 δ (x)2−a+p



in Ω .



• Any function w ∈ C2 (Ω ) ∩C(Ω ) that satisfies (4.67) also fulfills

w(x) ≤ c1 δ (x)



in Ω .



• Any function w ∈ C2 (Ω ) ∩C(Ω ) that satisfies

−Δ w ≥ δ (x)−a+p , w > 0



in Ω ,

on ∂ Ω ,



w=0

also has the property that

w(x) ≥ c2 δ (x)2−a+p



in Ω .



• Any function w ∈ C2 (Ω ) ∩C(Ω ) such that

−Δ w ≥ δ (x)−b+q(2−a+p) , w > 0



in Ω ,

on ∂ Ω ,



w=0

satisfies

w(x) ≥ c2 δ (x)



in Ω .



Now we fix M > 1 such that

min{M τ , M 1+(p+τ )q } > max{c1 , c−1

2 }.



4.3 Sublinear Lane–Emden Systems with Singular Data



161



We apply Schauder’s fixed point theorem for the mapping F defined by (4.70)

and (4.71) where this time the set A ⊂ C(Ω ) × C(Ω ) is given by



1





δ (x)2−a+p ≤u(x) ≤ M p+τ δ (x)2−a+p

p+τ

M

A = (u, v) ∈ C(Ω ) ×C(Ω ) :

1





δ (x) ≤v(x) ≤ M δ (x)

M







in Ω ⎪



in Ω ⎭



If a − p = 1 we fix 0 < ε < 1 and proceed in the same fashion with the set



1





δ (x) ≤ u(x) ≤ M p+τ δ (x)1−ε in Ω

p+τ

M

A = (u, v) ∈ C(Ω ) × C(Ω ) :

1





δ (x) ≤ v(x) ≤ M δ (x)

in Ω

M















.



,



where M > 1 is a suitably chosen constant. Finally, if a − p < 1 we define the set A

as



1





δ (x) ≤ u(x) ≤ M p+τ δ (x)

p+τ

M

A = (u, v) ∈ C(Ω ) × C(Ω ) :

1





δ (x) ≤ v(x) ≤ M δ (x)

M







in Ω ⎪



in Ω ⎭



.



For the uniqueness, we first remark that any solution (u, v) of (4.64) satisfies v ∈

C2 (Ω ) ∩C1 (Ω ).



Indeed, by Proposition 4.7 there exists c > 0 such that u, v ≥ cδ (x)



in Ω . Thus, −Δ v ≤ cδ (x)−b+q in Ω so, by Lemma 4.22(i)–(ii) we have v ∈ C1 (Ω )

and v(x) ≤ c0 δ (x) in Ω for some c0 > 0.

Let (u1 , v1 ) and (u2 , v2 ) be two solutions of (4.64). Using the above remark, there

exists 0 < m < 1 such that

mδ (x) ≤ vi (x) ≥



1

δ (x)

m



in Ω , i = 1, 2.



(4.72)



Therefore, we can find a constant C > 1 such that Cv1 ≥ v2 and Cv2 ≥ v1 in Ω .

We claim that v1 ≥ v2 in Ω . Supposing the contrary, let

M = inf{A > 1 : Av1 ≥ v2 in Ω }.

By our assumption, we have M > 1. From Mv1 ≥ v2 in Ω , it follows that −Δ u2 =



δ (x)−a v2p ≤ M p δ (x)−a v1p in Ω . Hence −Δ (M −p u2 ) ≤ δ (x)−a v1p = −Δ u1 in Ω ,

which yields M −p u2 ≤ u1 in Ω . Using the last inequality we have

−Δ v1 = δ (x)−b u1 ≤ M −pq δ (x)−b u2

q



q



in Ω .



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4 Singular Lane–Emden–Fowler Equations and Systems



It follows that −Δ (M pq v1 ) ≤ δ (x)−b u2 = −Δ v2 in Ω , which implies M pq v1 ≤ v2

q



in Ω . Further we obtain −Δ u2 = δ (x)−a v2p ≥ M p q δ (x)−a v1p in Ω . As before we

2



now derive M −p q u2 ≥ u1 in Ω . Finally we have

2



−Δ v1 = δ (x)−b uq1 ≥ M −p



2 q2



δ (x)−b uq2



in Ω .



It follows that

−Δ (M p

Thus, M



p2 q2



2 q2



v1 ) ≥ δ (x)−b u2 = −Δ v2

q



in Ω .



v1 ≥ v2 in Ω . Since 0 < p2 q2 < 1, the above inequality contradicts the



definition of M. Thus, v1 ≥ v2 in Ω and similarly we obtain v2 ≥ v1 in Ω . Hence

v1 ≡ v2 which yields u1 ≡ u2 . The proof of (ii) is similar.



4.3.3 Case p < 0 and q < 0

Theorem 4.25 Let p, q < 0 satisfy pq < 1 and assume one of the following conditions holds:

(i) b − q < 1 and a − p < 2.

(ii) a − p < 1 and b − q < 2.

Then, the system (4.64) has a unique solution.

Proof. The proof is similar to that for Theorem 4.24. We only point out the

differences. We fix τ > 0 such that (p − τ )q < 1.

(i) If 1 < a − p < 2 then we proceed as in the proof of Theorem 4.24 for the set

A ⊂ C(Ω ) × C(Ω ) given by



M p−τ δ (x)2−a+p ≤u(x) ≤ M −p+τ δ (x)2−a+p



A = (u, v) ∈ C(Ω ) ×C(Ω ) :

1



δ (x) ≤v(x) ≤ M δ (x)

M





in Ω ⎬



in Ω ⎭



If a − p = 1 we fix 0 < ε < 1 and proceed in the same way with the set





M p−τ δ (x) ≤ u(x) ≤ M −p+τ δ (x)1−ε in Ω ⎬



A = (u, v) ∈ C(Ω ) × C(Ω ) :

,

1



δ (x) ≤ v(x) ≤ M δ (x)

in Ω ⎭

M

where M > 1 is a suitably chosen constant. Finally, if a − p < 1 we define A as



.



4.3 Sublinear Lane–Emden Systems with Singular Data



163









M p−τ δ (x) ≤ u(x) ≤ M −p+τ δ (x)

A = (u, v) ∈ C(Ω ) × C(Ω ) :

1



δ (x) ≤ v(x) ≤ M δ (x)

M





in Ω ⎬

in Ω ⎭



.



For the uniqueness, we first remark that any solution (u, v) of (4.64) satisfies

c1 δ (x) ≤ v ≤ c2 δ (x) in Ω . Indeed, by Proposition 4.7 there exists c > 0 such that

u, v ≥ cδ (x) in Ω . Then −Δ v ≤ cq δ (x)−b+q in Ω so, according to Lemma 4.22(ii)

we have v ≤ c0 δ (x) in Ω for some c0 > 0.

Let (u1 , v1 ) and (u2 , v2 ) be two solutions of (4.64). By the above remark vi

(i = 1, 2), are both comparable to the distance function δ (x) up to the boundary.

Therefore, we can find a constant C > 1 such that Cv1 ≥ v2 and Cv2 ≥ v1 in Ω . We

next proceed in the same manner as in the proof of Theorem 4.24. The proof of (ii)

is similar.



4.3.4 Further Extensions: Superlinear Case

We want to point out here some features of the superlinear case p, q > 0 and pq > 1.

In this setting, system (4.64) has a variational structure. More precisely, one can see

(4.64) as a Hamiltonian system:



−Δ u = Hv (x, u, v)







−Δ v = Hu (x, u, v)







u=v=0

where

H(x, u, v) =



in Ω ,

in Ω ,

on ∂ Ω ,



u p+1

vq+1

+

.

(p + 1)δ (x)a (q + 1)δ (x)b



The approach in this case is variational, it consists of using the fractional powers

of the negative Laplace operator subject to homogeneous Dirichlet boundary conditions.

In fact, we deduce the existence of solutions for a more general system, namely



−Δ u = δ (x)−a |v| p−1v in Ω ,







(4.73)

−Δ v = δ (x)−b |u|q−1u in Ω ,







u=v=0

on ∂ Ω .

Our main result concerning system (4.73) is the following.



164



4 Singular Lane–Emden–Fowler Equations and Systems



Theorem 4.26 Let p, q > 0 satisfy pq > 1 and

1−a 1−b N−2

+

>

,

1+ p 1+q

N

p<



2N(1 − a)

N −4



q<



and



(4.74)



2N(1 − b)

N −4



if N ≥ 5.



(4.75)



Then, the system (4.73) has infinitely many solutions of which at least one is positive.

Proof. The proof is similar to that in [63, Theorem 1]; we only point out here the

main differences. Consider the Laplace operator

−Δ : H 2 (Ω ) ∩ H01 (Ω ) ⊂ L2 (Ω ) → L2 (Ω ),

and denote by {λn , en } the corresponding eigenvalues and eigenfunctions with

0 < λ1 ≤ λ2 ≤ · · · ≤ λn → ∞



and



en



2



= 1.



Thus, any u ∈ H 2 (Ω ) ∩ H01 (Ω ) has the unique representation

u=



∑ a n en



where an =



n≥1



Ω



uen dx.



For any 0 < s < 1 we define

E s = {u =



∑ an en ∈ L2 (Ω ) : ∑ λn2s a2n < ∞}.



n≥1



n≥1



The s-power As of −Δ is defined as

As : E s ⊂ L2 (Ω ) → L2 (Ω ),



As =



∑ λnsan en.



n≥1



It turns out that E s is a Hilbert space with the inner product

u, v



Es



=



Ω



As uAs vdx.



Moreover, E s is a fractional Sobolev space (see [135]) and E s ⊆ H 2s (Ω ) for all

0 < s < 1. Further, the embedding E s → Lr (Ω ) is compact provided



1

r



Note that by Hăolder inequality we have







uq+1

dx u

δ (x)b



q+1

Lr (Ω )



−br



Ω



δ (x) r−(q+1) dx



r−(q+1)

r



≤C u



q+1

,

Lr (Ω )







1

2



− 2s

N.



4.3 Sublinear Lane–Emden Systems with Singular Data



165



for all u ∈ E s and r > (q + 1)/(1 − b). Thus the embedding

E s → Lq+1 (Ω , δ (x)−b )

is compact. Using (4.74)–(4.75), one can find 0 < s,t < 1 such that s + t = 1 and

2N(1 − b)

.

1+q



2N(1 − a)

1+ p



and



N − 4s <



E s → Lq+1 (Ω , δ (x)−b )



and



E t → L p+1 (Ω , δ (x)−a )



N − 4t <

Then the embeddings



are compact.

Let E = E s × E t . We first look for (s,t)-weak solutions to (4.73) in the following

sense.

Definition 4.1. We say that (u, v) is an (s,t)-weak solution of system (4.73) if

Ω



As uAt φ dx +



Ω



At vAs ψ dx −



Ω



vp

φ dx −

δ (x)a



Ω



uq

ψ dx = 0,

δ (x)b



for all (φ , ψ ) ∈ E.

It is easy to see that any (s,t)-weak solution of (4.73) is in fact a critical point of

the functional

I : E → R,



I(u, v) =



Ω



As uAt vdx −



Ω



H(x, u, v)dx.



Remark that if (u, v) is an (s,t)-weak solution of (4.73) then

u ∈ W 2,p1 (Ω )



and



v ∈ W 2,q1 (Ω ),



and



q1 q +



for all p1 , q1 > 1 that satisfy

p1 p +



2Na

2N

<

N − 4t

N − 4s



2Nb

2N

<

.

N − 4s

N − 4s



From now on we employ step by step the same arguments as in [63] in order to

deduce that system (4.73) has infinitely many solutions of which at least one is

positive.







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