Shooting method with sign-changing nonlinearity

Nonlinear Analysis 114 (2015) 2–12

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Shooting method with sign-changing nonlinearity Ze Cheng a,b , Congming Li a,b,∗ a

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

b

Department of Applied Mathematics, University of Colorado Boulder, CO 80309, USA

article

abstract

info

Article history: Received 10 May 2014 Accepted 17 October 2014 Communicated by Enzo Mitidieri

In this paper, we study the existence of solution to a nonlinear system:

MSC: 35B09 35B33 35J30 35J48 53A30 34A34

for sign changing nonlinearities fi ’s. A degree theory approach to shooting method for this type of problems is introduced independently by Li (2013) and Liu et al. (2006) for nonnegative fi ’s. However, many systems of nonlinear Schödinger type involve interaction with undetermined sign. Here, based on some new dynamic estimates, we are able to extend the degree theory approach to systems with sign-changing source terms. © 2014 Elsevier Ltd. All rights reserved.



−∆ui = fi (u) ui > 0

in Rn , in Rn , i = 1, 2, . . . , L

(0.1)

Keywords: Existence of solution Degree theory Shooting method Rellich–Pohožaev identity

1. Introduction Consider the systems



−∆ui = fi (u) ui > 0

in Rn , in Rn ,

(1.1)

where i = 1, 2, . . . , L. Denote RL+ = {u ∈ RL |ui > 0, for i = 1, . . . , L}, and throughout this article f = (f1 , f2 , . . . , fL ) is assumed continuous in RL+ and locally Lipschitz continuous in RL+ . The existence and nonexistence of positive solution of this type of systems has long been studied. An important example (after reducing the order) is the Hardy–Littlewood–Sobolev type of system,

 (−△)k u = v p (−△)k v = uq  u, v > 0,

in Rn , in Rn ,

(1.2)

of which the Lane–Emden system is a special case (k = 1). One of the most interesting phenomena about the Lane–Emden system is that, there is a dividing curve of parameters (p, q) introduced independently by Clément–De Figueiredo– 1 Mitidieri [3] and Peletier–Van der Vorst [13,25], namely, p+ + q+1 1 = 1 − 2n , such that on or above it the Lane–Emden 1

∗

Corresponding author. E-mail address: [email protected] (C. Li).

http://dx.doi.org/10.1016/j.na.2014.10.019 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

3

system admits solutions and below it people conjecture that the system admits no solution (this is still open for n ≥ 5). 1 In [9,11] Mitidieri has proved the nonexistence for radial solution in the subcritical case, i.e. p+ + q+1 1 > 1 − 2n (see also 1 1 Serrin–Zou [20]). The critical case, p+ + q+1 1 = 1 − 2n , is known to exist solution by concentration compactness. In [22], 1 1 Serrin and Zou showed the existence of solution for supercritical case, p+ + q+1 1 < 1 − 2n . Thus, people move their focus 1 on the subcritical case. In [20] Serrin and Zou have proved a Liouville theorem in the space of n = 3 with an assumption of polynomial growth at infinity. Then in [16] Poláčik, Quittner and Souplet have removed that assumption and thus completes the proof of the case n = 3. In [23], Souplet has proved the Liouville theorem for n ≤ 4. For other related results for Lane–Emden system, see [2,4] and the reference within papers mentioned above. In [7] P.L. Lions has systematically developed the concept of concentration compactness. People find great application of this powerful tool in calculus of variations and nonlinear elliptic PDE, for example, to derive the existence of solution of 1 + q+1 1 = 1 − 2n . However, the argument no the Lane–Emden system in critical case, i.e. system (1.2) with k = 1 and p+ 1 longer applies for supercritical (p, q), and Serrin and Zou used shooting method to get the existence (see [22]). In [5], C. Li introduces a degree approach to shooting method to obtain existence of radial positive solution for system (1.1), which can be used to prove the existence of solution to both critical and supercritical cases of Lane–Emden systems in a uniform way. The degree approach relates the existence of solutions in whole (global) space to the non-existence to a corresponding (local) Dirichlet problem. Such kind of idea is implemented by many mathematicians, for instance earlier by Berestycki, Lions and Peletier [1], and recently, in [8] Liu, Guo and Zhang, also following this idea, have obtained an existence result for polyharmonic systems with positive nonlinear source terms and independently introduced the degree approach. For more development of degree approach to shooting method, see Li and Villavert [6,24]. Besides some non-degeneracy and growth control requirements, the conditions places on f of system (1.1) in previous mentioned works by Li [6], Liu–Guo–Zhang [8], Serrin–Zou [22,21], Villavert [24] all include that f needs to be positive (i.e. fi is positive for all i = 1, . . . , L). Although several classes of systems such as critical and supercritical cases of (1.2) and etc. are covered by those works, there are cases of nonlinear Shrödinger type of systems that f does not need to be always positive. In this article, we are going to replace the condition f being positive by fi ≥ 0 where each fi can change sign. Since we are looking for positive radial solution u(x) = u(|x|) of (1.1), the problem is equivalent to looking for global positive solution to the following ODE system,

n−1 ′ ui (r ) = −fi (u(r )) r ′ ui (0) = 0, ui (0) = αi for i = 1, 2, . . . , L.



u′′i (r ) +

(1.3)

where α = (α1 , α2 , . . . , αL ) is positive (i.e. each αi > 0) initial value for u. By classical ODE theory, this initial value problem (1.3) has a unique solution u(r , α) for r in some maximum interval. Let rα := infr ≥0 {r ∈ R|u(r , α) ∈ ∂ RL+ } (by ∂ RL+ we mean the boundary of RL+ , which sometimes we call ‘‘the wall’’), so r = rα is where u(r , α) touches the wall for the first time. There are two cases, case 1: rα = ∞, u never hits wall, then u(r , α) is a radial positive solution to (1.1), and we are done; case 2: rα < ∞, u hits wall in finite time, and we will show that the existence of solution of (1.1) will sufficiently determined by the non-existence of radial solution to its corresponding Dirichlet problem (1.4) in the below,

 −∆ui = fi (u) in BR , ui > 0 in BR , ui = 0 on ∂ BR , where BR = BR (0) for any R > 0 and i = 1, 2, . . . , L.

(1.4)

For local non-existence part, we follow Mitidieri’s work in [10] by implementing Rellich–Pohožaev (see Pohožaev [14] and Rellich [19]) type of identities, and for completeness we will give proof for some examples in Section 3. See also the pioneering work of Pohožaev [15] about Pohožaev type of identities for general variational problem. Here is our main result, Theorem 1.1. Given the nonexistence of radial solution to system (1.4) for all R > 0, the system (1.1) admits a radially symmetric solution of class C 2,α (Rn ) with 0 < α < 1, if f = (f1 (u), f2 (u), . . . , fL (u)) : RL → RL satisfies the following assumptions: 1. f is continuous in RL+ and locally Lipschitz continuous in RL+ , and furthermore, L 

fi (u) ≥ 0 in RL+ ;

(1.5)

i=1

2. If α ∈ ∂ RL+ and α ̸= 0, i.e., for some permutation (i1 , . . . , iL ), αi1 = · · · = αim = 0, αim+1 , . . . , αiL > 0 where m is an integer in (0, L), then ∃δ0 = δ0 (α) > 0 such that for β ∈ RL+ and |β − α| < δ0 , L  j=m+1

|fij (β)| ≤ C (α)

m 

fij (β),

j =1

where C is a non-negative constant that depends only on α .

(1.6)

4

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

Remark 1.2. Notice that under assumptions above, sign-changing source term f is allowed. It is known that if fi ’s stay positive, there are many nice properties that we can use to obtain existence results, for example, Serrin and Zou’s paper on Lane–Emden system [22] and on Hamiltonian type [21] and some recent work done by Li and Villavert [5,6,24] and Liu, Guo and Zhang’s work in [8]. If fi changes sign, those properties are lost, which leads to estimates failing. Our work here is to derive dynamic estimate (2.3) and (2.9) under assumptions above, such that the degree theory approach to shooting method is applicable to show existence of solution with sign-changing f . The paper is organized as follows. In Section 2, we prove the main result. In Section 3, we show nonexistence of solution to the corresponding Dirichlet problems of some example systems. 2. Proof of the main result In this section, we will first define a target map and prove its continuity. Then we apply degree theory to prove Theorem 1.1. 2.1. Target map For any real number a > 0, let Σa = {α ∈ RL+ | i=1 αi = a}, and Ba = {α ∈ ∂ RL+ | i=1 αi ≤ a}. Recall that for positive α (i.e. every αi > 0) we define rα = infr ≥0 {r ∈ R|u(r , α) ∈ ∂ RL+ }. As mentioned before, we can assume rα < ∞ since if rα = ∞ we get a solution to (1.1). Then we define a target map on a initial data as following,

L

L

Definition 2.1. Let u(r , α) be a solution to (1.3) with initial value α ∈ Σa , we define a map ψ : Σa → ∂ RL+ , such that

ψ(α) =

u(rα , α)



α

α ∈ RL+ , α ∈ ∂ RL+ .

(2.1)

Here we sketch Li’s degree theory approach to shooting method as follows. Fix any real number a > 0, and assume that for any initial value α ∈ Σa no global positive solution to (1.3) exists (i.e. rα < ∞), so we can define a target map. Hence, step 1, we show that, under some suitable assumptions on f , the target map ψ is continuous from Σa to Ba ; step 2, by degree theory we show that ψ is onto, therefore ∃α0 ∈ Σa such that ψ(α0 ) = u(rα0 , α0 ) = 0; step 3, note that by assumption rα0 < ∞, u(r , α0 ) for r ∈ [0, rα0 ] is a solution to the Dirichlet problem (1.4) with R = rα0 , which makes a contradiction if we assume that system (1.4) admits no radial solution for any R > 0. In what follows, we assume (1.3) admits no global positive solution, i.e. rα < ∞. We first show that under our assumptions (1.5) and (1.6) on f , the behavior of u turns regular, such that ψ is continuous. Lemma 2.2. For any real number a > 0, let

 Σa =

α ∈ RL+ |

L 





αi = a

Ba =

and

α ∈ ∂ R+ |

i =1

L

L 

 αi ≤ a .

i=1

The target map ψ defined in Definition 2.1 is a continuous map from Σa to Ba if f satisfies assumptions (1.5) and (1.6). Proof. To see that ψ maps Σa to Ba , we need to notice that by assumption (1.5) ΣiL=1 fi ≥ 0, so we solve from the ODE system (1.3) and get

ΣiL=1 ui (r , α) = ΣiL=1 αi − ΣiL=1

 r  s  n−1 τ 0

0

s

fi (u(τ ))dτ ds

≤ ΣiL=1 αi , for r ∈ [0, rα ]. Therefore, ψ(α) ∈ Ba . Next, we will show that ψ is continuous on Σa . Fix any α ∈ Σa , then α lies on the boundary of RL+ or in RL+ (α ̸= 0 since a > 0), and we will prove for these two cases. Case 1. α ∈ ∂ RL+ . Suppose α i1 = · · · = α im = 0, and α im+1 , . . . , α iL > 0, for some integer m that 0 < m < L. By the second assumption (1.6), ∃δ0 > 0, such that for α ∈ RL+ satisfying |α − α| < δ0 we have L  j=m+1

|fij (α)| ≤ C (α)

m 

fij (α).

(2.2)

j=1

Notice that we can choose C = C (α) ≥ 1 in (2.2). First, we claim that 1≤j≤m fij (α) ≥ 0. Indeed, if C = 0, the term on the   left of the inequality (2.2) must be zero, and due to the first assumption (1.5), 1≤j≤m fij (α) ≥ 0. If C > 0, then 1≤j≤m fij (α)  ≥ 0 obviously. Since 1≤j≤m fij (α) ≥ 0 we choose C ≥ 1.



Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

5

For this δ0 and C , we claim that: δ Fix any δ < 2(3+0L)C , and for α ∈ Σa such that |α − α| ≤ δ , then for r ∈ [0, rα ]

|u(r , α) − α| ≤ 2(3 + L)C δ < δ0 .

(2.3)

As we will see in the following proof, (2.3) is a dynamic estimate, in the sense that if |u(r , α) − α| < δ0 with r ∈ [0, a1 ) ⊂

[0, rα ] for some a1 > 0, then by (1.6) we have L 



|fij (u(r , α))| ≤ C (α)

j=m+1

fij (u(r , α)),

(2.4)

1≤j≤m

and this control enable us to push the range of r in |u(r , α) − α| < δ0 further than a1 and up to rα . Suppose the claim not true, then there exists α0 ∈ RL+ satisfying |α0 − α| ≤ δ and a1 ∈ (0, rα0 ) such that the equality of (2.3) holds at r = a1 for the first time, i.e.

|u(r , α0 ) − α|



< 2(3 + L)C δ, = 2(3 + L)C δ,

if r < a1 , if r = a1 .

(2.5)

For r ∈ (0, a1 ) we have,

|u(r , α0 ) − α| ≤ |u(r , α0 ) − α0 | + |α0 − α| ≤

m 

|uij (r , α0 ) − α0ij | +

(2.6)

L 

|uij (r , α0 ) − α0ij | + |α0 − α|.

(2.7)

j=m+1

j =1

So, to estimate the second term of (2.7), we solve from (1.3) and get L 

|uij (r , α0 ) − α0ij | =

j=m+1

  r  s   L    τ n −1   f ( u (τ )) d τ ds ij   s

j=m+1

0

0

Notice that since C ≥ 1, 2(3 + L)C δ < δ0 , therefore assumption (1.6) can be applied on u(r , α0 ) for r ∈ (0, a1 ), L 

  r  s  n−1 L   τ    dτ ds f ( u (τ )) ij  s 

|uij (r , α0 ) − α0ij | ≤

j=m+1

j=m+1

≤C

0

=C

0

0

 r  s  n−1  m τ 0

s

fij (u(τ ))dτ ds

j =1

m  (α0ij − uij (r , α0 )), j =1

The first term of (2.7) can be estimated by m 

|uij (r , α0 ) − α0ij | =

j=1

m m   (α0ij − uij (r , α0 ))+ + (α0ij − uij (r , α0 ))− j =1

j =1

m  ≤2 (α0ij − uij (r , α0 ))+ . j=1

To see the inequality above, one needs to notice that due to (1.6), on [0, rα0 ]. So,

m

j=1 fij

(u) ≥ 0, so

m

j =1

uij (r , α0 ) is monotone decreasing

m m m    (α0ij − uij (r , α0 )) = (α0ij − uij (r , α0 ))+ − (α0ij − uij (r , α0 ))− ≥ 0. j =1

j =1

j =1

The last term of (2.7) is bounded by δ , and notice that ui (r , α0 ) > 0, i = 1, . . . , L for r ∈ (0, rα0 ), so we get

|u(r , α0 ) − α| ≤ 2

m m   (α0ij − uij (r , α0 ))+ + C (α0ij − uij (r , α0 )) + δ j =1

≤ (2 + C )

j =1 m  j=1

α0ij + δ

6

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

≤ (2 + C )L|α0 − α| + δ ≤ (3 + C )Lδ, where C is the same C in (1.6). Then we get a contradiction with (2.5) by taking r = a1 in the above. Hence the claim is proved. Notice that the claim and estimate (2.3) implies the continuity of ψ at α , therefore, we have proved case 1. Case 2. α ∈ RL+ . As above we assume rα < ∞, and obviously rα > 0. Let us assume at r = rα , for some integer m that 0 < m ≤ L (m > 0 because ψ(α) = u(rα , α) ∈ ∂ RL+ , i.e. u touches wall at r = rα ), so suppose ui1 (rα , α) = · · · = uim (rα , α) = 0, and m ′ uim+1 (rα , α), . . . , uiL (rα , α) > 0. Let ω(r ) = j=1 uij (r , α), and we claim that ω (rα ) < 0. By the continuity of u(r , α) with respect to r , ∃δ1 > 0 such that for r ∈ (rα − δ1 , rα ],

|u(r , α) − ψ(α)| < δ0 .

(2.8)

If m < L, the assumption (1.6) is taking effect, and therefore assumption (1.5), we also have

−

1

r

(r n−1 ω′ (r ))′ = n−1

m

j=1 fij

≥ 0 for r ∈ (rα − δ1 , rα ]; if m = L then by the

j=1 fij ≥ 0 for r ∈ (rα − δ1 , rα ]. So, in (rα − δ1 , rα ]

m

m 

fij ≥ 0.

(2.9)

j =1

Also, since ω(r ) > 0 for r < rα and ω(rα ) = 0, there must exist r0 ∈ (rα − δ1 , rα ), such that ω′ (r0 ) < 0. So, for r ∈ [r0 , rα ],

ω ′ (r ) =

 r n−1 0

r

ω′ (r0 ) −

 r  n−1  m τ r0

r

fij (u(τ ))dτ < 0.

(2.10)

j =1

This proves the claim ω′ (rα ) < 0. Then there exists l0 ∈ {1, . . . , m} such that u′il (rα , α) < 0. Therefore, combining with the 0

fact uil (rα , α) = 0, we see uil crosses the wall with a non-zero slope, i.e. there is a transversality at r = rα , and by classical 0 0 ODE stability theory (the continuous dependence on initial value) ψ is continuous at α .  2.2. Application of degree theory

Now, we recall some results in degree theory (in particular, the treatment modified by P. Lax, cf. [12]). Consider C ∞ oriented manifolds X0 , Y of dimension n (all manifolds are assumed to be paracompact) and an open subset X ⊆ X0 with compact closure. For convenience, write dy1 ∧ · · · ∧ dyn = dy. Then for a C 1 map φ : X → Y , the degree is defined as following: Definition 2.3. Let µ = f (y)dy be a C ∞ n-form with support contained in a coordinate patch Ω of y0 and lying in Y \{φ(∂ X )} such that Y µ = 1; set deg(φ, X , y0 ) =



µ ◦ φ.

(2.11)

X

Here are some properties of degree which we will refer to in our proof, Proposition 2.4. For y1 close to y0 , deg(φ, X , y0 ) = deg(φ, X , y1 ). It follows that the degree of a mapping is constant on any connected component C of Y \ {φ(∂ X )}, and we can write degree as deg (φ, X , C ). Proposition 2.5. If y0 ̸∈ φ(X ), then deg(φ, X , y0 ) = 0. As a result, if deg(φ, X , y0 ) ̸= 0, then y0 ∈ φ(X ). An important property of degree is that, the notion can be extended to maps φ which are merely continuous, since we can approximate such φ by C 1 maps φn (see Property 1.5. in [12]). Also, degree is homotopy invariant (see Proposition 1.4.3 in [12]), which enables us to define degree for continuous map η : ∂ X → Rn \ y0 (see Property 1.5.4 in [12]). It leads to the following theorem (see Property 1.5.5 in [12]), Theorem 2.6. deg(η, X , y0 ) depends only on the homotopy class of η : ∂ X → Y = Rn . The above theorem is also true if we change Rn to a hyperplane T n = α ∈ Rn |Σin=1 αi = a , i.e.,



Theorem 2.7. deg(η, X , y0 ) depends only on the homotopy class of η : ∂ X → Y = T n . Now we are prepared to prove the existence of solution to (1.1).



Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

7

Proof of Theorem 1.1. For any fixed real number a > 0, assume that (1.3) admits no global positive solution with any initial value α ∈ Σa , so rα < ∞, and then we can define a target map ψ by (2.1). Recall that Σa = {α ∈ RL+ | i=1 αi = a} and Ba = {α ∈ ∂ RL+ | i=1,...,L αi ≤ a}, and by Lemma 2.2 ψ is a continuous maps from Σa −→ Ba .  Let π (α) = α + 1L (a − i=1,...,L αi )(1, . . . , 1) : Ba −→ Σa , then π is continuous with a continuous inverse π −1 (α) = α − (mini=1,...,L αi )(1, . . . , 1) : Σa −→ Ba . The map: φ = π ◦ ψ : Σa −→ Σa is continuous and φ(α) = α on ∂ Σa . Then let η = id (the identity map) and for n = L − 1, X = Σa \ ∂ Σa , Y = T n and then by Theorem 2.7 we have deg(φ, X , α) = deg(η, X , α) = 1 for any α ∈ X . By Property 2.5, φ is onto, which implies that ψ is also onto. This shows that there exists an α0 ∈ Σa such that ψ(α0 ) = 0. Since we assume that system (1.4) admits no solution, rα0 corresponding to this α0 cannot be finite, a contradiction. This completes the proof of Theorem 1.1. 

L



3. Examples One of the simplest systems is that f ≡ 0 in (1.1), then u ≡ C for some constant vector C is a solution. Let us point out that since f is not positive, this trivial system is not included in results of Li and Villavert [5,6,24], but it is included in our cases. In this section, we will show the existence of solution to some non-trivial systems. In the view of Theorem 1.1, we only need to show their corresponding Dirichlet problems admit no radial solution, and verify its source term satisfy our assumptions in Theorem 1.1. The main tool of showing non-existence of solution (hence no radial solution) to Dirichlet problem is Rellich–Pohožaev type of identities. Mitidieri did the pioneering work in [10], and here we present a brief proof for completeness (see also Quittner–Souplet [18]). Here the Rellich–Pohožaev identities we need are the following, Lemma 3.1. For the following system,

 −∆u = f (u, v)   −∆v = g (u, v)  u, v > 0  u, v = 0

in B, in B, in B, on ∂ B,

in Rn ,

(3.1)

we have (i) for θ ∈ [0, 1],





θ v ∆u + (1 − θ )u∆v dx,

∇ u · ∇v dx = − B

(3.2)

B

(ii)



∆u(x · ∇v) + ∆v(x · ∇ u) − (n − 2)∇ u · ∇v dx = B

 ∂B

(x · ν)



∂ u ∂v ∂ν ∂ν



dσ ≥ 0,

(3.3)

where ν is the outward normal, (iii)



∆u(x · ∇ u)dx =



B

n−2 B

2

2

|∇ u| dx +

1



2 ∂B

x · ν|∇ u|2 dσ ≥ 0.

(3.4)

Proof. Suppose there exists a positive solution (u, v) to (3.1). (i) Since

 −

v ∆udx =



B



u∆v dx,

∇ u · ∇v dx = − B

B

for θ ∈ [0, 1] we have





θ v ∆u + (1 − θ )u∆v dx.

∇ u · ∇v dx = − B

B

(ii)

  ∂u ∂v ∆u(x · ∇v) + ∆v(x · ∇ u)dx = (x · ∇v) + (x · ∇ u)dσ − (2∇ u · ∇v + xj ∂i u∂ij2 v + xj ∂i v∂ij2 u)dx ∂ν B ∂ B ∂ν B   ∂u ∂v = (x · ∇v) + (x · ∇ u)dσ − (2∇ u · ∇v + x · ∇(∇ u · ∇v))dx ∂ν ∂ν ∂ B B   ∂u ∂v = (x · ∇v) + (x · ∇ u)dσ − (2∇ u · ∇v dx − x · ν(∇ u · ∇v))dσ + n ∇ u · ∇v dx. ∂ν B ∂B B ∂ B ∂ν



8

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

∂u ∂v Notice the fact that x = |x|ν , and ∇ u = ∂ν ν and ∇v = ∂ν ν due to u, v = 0 on ∂ B, and after rearrangement we have ∂u ∂v the identity (3.3). Also, ∂ν ≤ 0 and ∂ν ≤ 0 on ∂ B, so we have RHS of (3.3) ≥ 0. (iii) If we let u = v , then (ii) gives



∆u(x · ∇ u)dx =



B

n−2 2

B

2

|∇ u| dx +

1



2 ∂B

x · ν|∇ u|2 dσ ≥ 0. 

3.1. Sign-changing source terms First we give a simple example that can be easily generated to some non-trivial sign-changing source terms systems. Consider the following system,

 −∆ u = v p − u p , −∆ v = u p , u, v > 0 ,

in Rn ,

(3.5)

and its corresponding Dirichlet problem,

 −∆u = v p − up   −∆v = up   u, v > 0 u, v = 0

in B, in B, in B, on ∂ B,

(3.6)

where B = BR (0) ⊂ Rn for any R > 0. We have 2 , then (3.5) admits radial positive solution. Theorem 3.2. If p > nn+ −2

Again the proof relies on the non-existence of solution to (3.6). 2 Lemma 3.3. If p > nn+ , then system (3.6) admits no solution for any R > 0. −2

Proof. By Lemma 3.1, we merge the source terms into (3.3) (we replace −up in the first source term by ∆v , i.e. −∆u = v p + ∆v and −∆v = up ), and by (3.2) we have



−(v p + ∆v)(x · ∇v) − up (x · ∇ u) − (n − 2)∇ u · ∇v dx  p+1   v up+1 = −x · ∇ + − ∆v(x · ∇v) + (n − 2) (θ v ∆u + (1 − θ )u∆v) dx p+1 p+1 B  p+1     v up+1 = −x · ∇ + − ∆v(x · ∇v) + (n − 2) −θ (v p+1 + v ∆v) − (1 − θ )up+1 dx p + 1 p + 1 B       n n − (n − 2)θ v p+1 + − (n − 2)(1 − θ ) up+1 dx = p+1 p+1 B  − ∆v(x · ∇v) + (n − 2)θ v ∆v dx.

LHS of (3.3) =

B

B

So, by (3.4), (3.3) becomes

 

  p+1 − (n − 2)θ v + − (n − 2)(1 − θ ) u dx p+1 p+1 B     ∂ u ∂v = ∆v(x · ∇v) + (n − 2)θ v ∆v dx + (x · ν) dσ ∂ν ∂ν B ∂B       n−2 1 ∂ u ∂v = |∇v|2 dx + x · ν|∇ u|2 dσ − θ (n − 2)|∇v|2 dx + (x · ν) dσ 2 2 ∂B ∂ν ∂ν B B ∂B      n−2 1 ∂ u ∂v = (1 − 2θ ) |∇v|2 dx + x · ν|∇ u|2 dσ + (x · ν) dσ . 2 2 ∂ν ∂ν B ∂B ∂B

Take θ =

1 , 2

  B



n

p+1



n

and we have n

p+1

−

n−2 2



 p+1  1 v + up+1 dx =



2 ∂B

x · ν|∇ u|2 dσ +

 ∂B

(x · ν)

+2 So, by assumption p > nn− the LHS of the above identity <0, a contradiction. 2





∂ u ∂v ∂ν ∂ν



dσ ≥ 0.

(3.7)

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

9

Proof of Theorem 3.2. Directly we see that (3.5) satisfies our first main assumption (1.5). To see (1.6) is satisfied, just notice that if u = 0 then |f2 | = 0 ≤ f1 = v p , and if v = 0 then |f1 | = up ≤ f2 . Then for a neighborhood of such (u, v) (i.e. uv = 0), (1.6) holds. So, combined with Lemma 3.3, Theorem 3.2 follows from Theorem 1.1.  Remark 3.4. We thank the referees for pointing out that, actually system (3.5) can be solved as follows. Suppose −∆w = w p , let u = λw and v = νw , then we can find suitable λ, ν such that (u, v) is a solution. However, consider a similar system,

 −∆ u = v p + v q − u p , −∆ v = u p , u, v > 0,

in Rn ,

(3.8)

2 . This system cannot be solved trivially as above, but we can show the existence of solution where p ̸= q and p, q > nn+ −2 to it by Theorem 1.1. We only need to show the nonexistence of solution to corresponding Dirichlet problem (to show source terms satisfying assumptions (1.5) and (1.6) is similar to the proof in Theorem 3.2). By Lemma 3.1, we merge −∆u = v p + v q + ∆v and −∆v = up into (3.3), and take θ = 12 then we have an identity similar to (3.7),

 

n p+1

B

=

1



2 ∂B

−

n−2



2

 p+1  v + up+1 +

x · ν|∇ u|2 dσ +

 ∂B

(x · ν)





n q+1

∂ u ∂v ∂ν ∂ν



−

n−2 2



v q+1 dx

dσ > 0.

(3.9)

+2 Then for p, q > nn− and p ̸= q, the above equation cannot hold and therefore the nonexistence of solution to the Dirichlet 2 problem follows.

3.2. Conservative source terms In this section, we consider a system that f has a potential function F , i.e. f = ∇ F , and F (0) = 0. Type I. Consider the following system, s

 ∂F −∆ u i = , ∂ ui ui > 0,

in Rn ,

(3.10)

where i = 1, . . . , L. The corresponding Dirichlet problem is

 ∂F   −∆ u i = ∂ ui   ui > 0 ui = 0

in B, (3.11)

in B, on ∂ B,

where i = 1, . . . , L, and B = BR (0) ⊂ Rn for any R > 0. In [17], Pucci and Serrin have showed that for a general variational problem,

 Ω

F (x, u, Du)dx = 0,

there exists Pohožaev type of identity that can give a sufficient condition on the nonexistence of solution to Dirichlet problem on a bounded star-shaped domain. In this case, the system (3.10) corresponds to a vector-valued extremal of the variational problem with

F (x, u, Du) =

L 1

2 k=1

|pk |2 − F (u),

where pk = Duk . So, Theorem 6 in Pucci–Serrin [17] leads to the following result. Lemma 3.5. For u ∈ RL+ , F is a C 1 function of u and satisfies F (0) = 0 and n−2 2

uk

∂F − nF (u) > 0, ∂ uk

for u ̸= 0,

then system (3.11) admits no nontrivial solution.

(3.12)

10

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

Proof. Suppose there exists a solution u = (u1 , . . . , uL ) to system (3.11). By Lemma 3.1, for i = 1, . . . , L,



∆ui (x · ∇ ui )dx −



B

n−2 2

B

2

|∇ ui | dx =

1



2 ∂B

x · ν|∇ ui |2 dσ ≥ 0.

Sum the LHS of the above identities up and get

∂F n−2 ( x · ∇ ui ) + ui ∆ui dx ∂ u 2 i B n − 2 ∂F = −x · ∇ F (u(x)) − dx ui 2 ∂ ui B n − 2 ∂F ui dx, = nF − 2 ∂ ui B 

0 ≤

−

which contradicts to (3.12).



Therefore, it follows from Theorem 1.1 and Lemma 3.5 that Theorem 3.6. For system (3.10), if f satisfies the assumptions (1.5) and (1.6), and additionally if f = ∇ F , where F (0) = 0, and F satisfies (3.12) for u ̸= 0, then system (3.10) admits a radially symmetric solution of class C 2 (Rn ). Here is an example system of Type I: Let the potential function be F (u, v) = −(u − v)2 + v p u + uq v,

with p, q >

n+2 n−2

,

(3.13)

and then

 −∆u = Fu = −2(u − v) + v p + quq−1 v, −∆v = Fv = 2(u − v) + pv p−1 u + uq ,  u, v > 0,

in Rn .

(3.14)

We can verify that F satisfies the condition of Theorem 3.6. F (0, 0) = 0, and Fu + Fv ≥ 0 ((1.5) is satisfied). Let u = 0 then Fu = 2v + v p ≥ |Fv | = 2v , similarly if v = 0, then Fv = 2u + uq ≥ |Fu | = 2u, so we can see that (1.6) is satisfied in a neighborhood. Last, direct computation shows that n−2 2

(uFu + v Fv ) − nF =

n−2

(−2(u − v)2 + (p + 1)uv p + (q + 1)uq v) − n(−(u − v)2 + v p u + uq v)     n−2 n−2 ≥ 2(u − v)2 + (p + 1) − n uv p + (q + 1) − n v uq 2

2

> 0,

2

for (u, v) ̸= (0, 0),

so (3.12) is satisfied. Also, notice that Fu and Fv are sign-changing functions, for example, let v = 0, then Fu < 0 and let u = 0 then Fu > 0. Type II. In the following example, we give another class of systems with potential type of source terms. Consider the following system, in Rn ,

 ∂F  −∆u = f1 = ,    ∂v ∂F  −∆v = f2 = ,   ∂u  u, v > 0,

(3.15)

and corresponding Dirichlet problem

 ∂F  −∆u = ,    ∂v   ∂F −∆v = ,  ∂u     u, v > 0, u, v = 0,

in B in B

(3.16)

in B on ∂ B

where B = BR (0) ⊂ Rn for any R > 0. Similarly we have the following local-nonexistence lemma obtained by Mitidieri in [10],

Z. Cheng, C. Li / Nonlinear Analysis 114 (2015) 2–12

11

Lemma 3.7. Let F be a C 1 function of u, v and satisfies F (0) = 0. In addition, if for (u, v) ̸= (0, 0), there exists θ ∈ [0, 1] such that

∂F ∂F (n − 2) θ u + (1 − θ )v ∂u ∂v 



− nF (u, v) > 0,

(3.17)

then system (3.16) admits no nontrivial solution. Then it follows from Theorem 1.1 and Lemma 3.7 that Theorem 3.8. For system (3.15), if f1 = Fv and f2 = Fu satisfies the assumptions (1.5) and (1.6), and additionally if F (0) = 0, and F satisfies (3.17) for (u, v) ̸= (0, 0), then system (3.15) admits a radially symmetric solution of class C 2 (Rn ). Proof of Lemma 3.7. Suppose there exists a solution (u, v). By Lemma 3.1, we merge the source terms to (3.3) and get,



∆u(x · ∇v) + ∆v(x · ∇ u) − (n − 2)∇ u · ∇v dx

0 ≤ B



−Fv (x · ∇v) − Fu (x · ∇ u) + (n − 2)(θ v ∆u + (1 − θ )u∆v)dx

= B = B =

−x · ∇ F − (n − 2)(θ v Fv + (1 − θ )uFu )dx nF − (n − 2)(θ v Fv + (1 − θ )uFu )dx,

B

which contradicts to (3.17).



Here is an example system of Type II: Let the potential function be F (u, v) =

|u − v|r +1 uq + 1 v p+1 + + , r +1 q+1 p+1

with p, q, r >

n+2 n−2

,

(3.18)

and

 −∆u = Fv = −|u − v|r −1 (u − v) + v p , −∆v = Fu = |u − v|r −1 (u − v) + uq ,  u, v > 0,

in Rn .

(3.19)

We verify that F satisfies conditions in Theorem 3.8. F (0) = 0, and Fu + Fv ≥ 0 ((1.5) is satisfied). Let u = 0 then Fv = v r + v p ≥ |Fu | = v r , similarly if v = 0, then Fu = ur + up ≥ |Fv | = ur , so we can see that (1.6) is satisfied in a neighborhood. Last, direct computation shows that by taking θ = 21 , n−2 2

  |u − v|r +1 uq + 1 v p+1 (|u − v|r +1 + uq+1 + v p+1 ) − n + + 2 r +1 q+1 p+1       n−2 n n − 2 n n − 2 n r +1 q +1 − |u − v| + − u + − v p+1 = 2 r +1 2 q+1 2 p+1 > 0, for (u, v) ̸= (0, 0),

(uFu + v Fv ) − nF =

n−2

so (3.17) is satisfied. Also, notice that Fu and Fv are sign-changing functions, for example, let v = 0, then Fv < 0 for u ̸= 0, and let u = 0 then Fv > 0 for v ̸= 0. Acknowledgments We thank the referees very much for many useful suggestions and constructive criticism which greatly improved both the result and the presentation of this paper. This work was done when Ze Cheng visited Shanghai Jiao Tong University, and we really appreciate the hospitality and support of SJTU. Ze Cheng is partially supported by NSF DMS-1405175. Congming Li is partially supported by NSFC-11271166 and NSF DMS-1405175 References [1] H. Berestycki, P.L. Lions, L.A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in RN , Indiana Univ. Math. J. 30 (1) (1981) 141–157. [2] W. Chen, C. Li, An integral system and the Lane–Emden conjecture, Discrete Contin. Dyn. Syst. 4 (24) (2009) 1167–1184.

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