Sharp criteria of Liouville type for some nonlinear systems

In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.


Introduction
In this paper, we establish sharp criteria for existence and nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations 1) and the corresponding nonlinear differential systems of Lane-Emden type equations (−∆) k u = v q , u, v > 0, (−∆) k v = u p , p, q > 0. (1.2) These systems are the 'blow up' equations for a large class of systems of nonlinear equations arising from geometric analysis, fluid dynamics, and other physical sciences. The nonexistence of positive solutions for systems of 'blow up' type like (1.1) and (1.2), known as Liouville type theorem, is useful in deriving existence, a priori estimate, regularity and asymptotic analysis of solutions. Another important topic is the study of the Wolff type system of nonlinear equations: (1.3) and the corresponding system of γ-Laplace equations Recall a Liouville-type theorem for the Lane-Emden equation −∆u = u p , in R n (n ≥ 3) (1.5) obtained by Caffarelli, Gidas and Spruck [1]: if p ∈ (0, n+2 n−2 ), then (1.5) has no positive classical solution. When p ≥ n+2 n−2 , (1.5) has positive classical solution. Namely, the right end point n+2 n−2 is a sharp criterion distinguishing the existence and the nonexistence. Numbers like this, separating the existence and the nonexistence, are called the critical exponents.
In the following theorem, we obtain the sharp criteria on the existence and the nonexistence of solutions to (1. 1. When k = 1, the first result is coincident with the result in [1]. 2. Part 2 of our Theorem 1.1 together with the nonexistence result of Souplet [48] imply that: for n ≤ 4, (1.2) has a pair of solutions if and only if 1 p+1 + 1 q+1 ≤ n−2k n .
Next, we consider the HLS type system of nonlinear equations (1.1) and its scalar case v ≡ u, q = p: (1.7) Such equations are related to the study of the best constant of Hardy-Littlewood-Soblev (HLS) inequality. Lieb [35] classified all the extremal solutions of (1.7), and thus obtained the best constant in the HLS inequalities. He posed the classification of all the solutions of (1. 7) as an open problem.
The corresponding PDE is the semilinear equation involving a fractional order differential operator (−∆) α/2 u = u (n+α)/(n−α) , u > 0, in R n . (1.8) The classification of the solutions of (1.8) with α = 2 has provided an important ingredient in the study of the prescribing scalar curvature problem. It is also essential in deriving priori estimates in many related nonlinear elliptic equations. It was well studied by Gidas, Ni, and Nirenberg [14]. They proved that all the positive solutions with reasonable behavior at infinity, namely are radially symmetric about some point. Caffarelli, Gidas, and Spruck removed the decay condition (1.9) and obtained the same result (cf. [1]). Then Chen and Li [5], and Li [28] simplified their proofs. Later, Chang, Yang and Lin also considered some higher order equations (cf. [3], [36]). Wei and Xu [50] generalized this result to the solutions of more general equation (1.8) with α being any even numbers between 0 and n. Chen, Li, and Ou solved the open problem as stated for the integral equation (1.7) or the corresponding PDE (1.8) in [11]. The unique class of solutions can assume the form (1. 10) water waves, molecular dynamics, relativistic quantum mechanics, and stable Levy process. The equivalence provides a technique in studying the PDEs: one can use the corresponding integral equations to investigate the global properties for those phenomena. A positive solution u of (1.7) is called a finite energy solution, if u ∈ L p+1 (R n ). Similarly, positive solutions u, v are called finite energy solutions of (1.1), if u ∈ L p+1 (R n ), v ∈ L q+1 (R n ). Now, we point out the relation between the critical conditions and the existence of finite energy solutions of (1.7) and (1.1). (1.14) Remark 1.2.
1. In the subcritical case p < n+α n−α , Theorem 3 in [9] shows that (1.7) has no locally finite energy solution by using the method of moving planes and the Kelvin transformation. For system (1.1), the proof of nonexistence in the subcritical case 1 p+1 + 1 q+1 > n−α n is rather difficult. It is usually called the HLS conjecture (cf. [2] and [6]). Partial results are known.
2. In the supercritical case p > n+α n−α with α = 2, Li, Ni and Serrin proved the semilinear Lane-Emden equation (1.5) has the decay solution (cf. [32], [40] and [42]). According to Corollary 1.3, the energy of those solutions u are infinite. Namely, u p+1 = ∞. Similarly, the positive solutions u, v obtained in [46] when 1 p+1 + 1 q+1 < n−2 n are not the finite energy solutions (i.e. u p+1 = v q+1 = ∞). The following γ-Laplace equation is also concerned in this paper (1. To study the γ-Laplace equations, we introduce the Wolff potential of a positive locally integrable function f The integral equation involving the Wolff potential is related with the study of many nonlinear problems. The Wolff potentials are helpful to understand the nonlinear PDEs such as the γ-Laplace equation and the k-Hessian equation (cf. [23], [25], [26] and [44]). According to [21], if inf R n u = 0, there exists C > 0 such that the positive solution u of (1.15) Thus, u solves (1.16) for some double bounded c(x). Here, a function c(x) is called double bounded, if there exists a positive constant C > 0 such that For the coupling system Chen and Li [7] proved the radial symmetry for the integrable solutions. Afterward, Ma, Chen and Li [38] used the regularity lifting lemmas to obtain the optimal integrability and the Lipschitz continuity. Based on these results, [24] obtained the decay rates of the integrable solutions when |x| → ∞.
The critical exponents and the critical conditions play a key role under the scaling transform. In the following, some interesting observations are listed.
Under the scaling transform u µ (x) = µ σ u(µx), Remark 1.4. Here an interesting observation is, the critical exponent n+γ n−γ (γ− 1) is different from the divided exponent γ * −1 in Theorem 1.4 except γ = 2. The reason is that those critical numbers are in the different finite energy functions classes L p+γ−1 (R n ) and L p+1 (R n ), respectively.
Next, we are concerned with the sufficient and necessary conditions for the existence of the positive solutions of equations and systems with some double bounded coefficients. Here, new divided numbers and conditions appear.
has positive solutions for some double bounded c(x) if and only if p > n n−α .
(1) Assume p > 1. The 2k-order PDE 20) has positive solutions for some double bounded c(x) if and only if p > n n−2k . (2) Assume pq > 1. The system has positive solutions u, v for some double bounded c 1 (x) and has positive solutions for some double bounded c(x), if and only if p > n(γ − 1) n − βγ .
(2) The system has positive solutions u, v for some double bounded c 1 (x) and c 2 (x), if and only if pq > (γ − 1) 2 and has positive solutions for some double bounded c(x). If 0 < p ≤ n(γ−1) n−γ , then for any double bounded c(x), (1.23) has no positive solution satisfying inf R n u = 0.
for some double bounded c 1 (x) and c 2 (x). On the contrary, for any double bounded functions c 1 (x) and c 2 (x), if one of the following conditions holds Then (1.24) has no positive solutions u, v satisfying inf R n u = inf R n v = 0. In the proofs of Theorem 1.5-Corollary 1.8, we apply a special iteration scheme and some critical asymptotic analysis to establish the existence and the nonexistence, and hence obtain the sharp criteria.

Equations with variable coefficients
The following proposition is often used in this paper.
Proof. In view of w L 1 (R n ) < ∞, it follows from the definition of the improper integral that Hence, as R → ∞, There exist R j ∈ [R, 2R], such that as R j → ∞, |w|ds → 0 and R n j S n−1 |w|ds → 0.

HLS type integral equation
In this subsection, we give a relation between the exponents and the existence of positive solutions for integral equations involving the Riesz potentials. First we consider the semilinear Lane-Emden type equations where θ > 0 will be determined later. Denote |x| by r, and set U (r) = U (|x|) = u(x). By a simply calculation, we obtain Namely, (2.2) with the slow rate 2θ = 2 p−1 is a solution for some double bounded c(r).
Moreover, if p = n+2 n−2 , there also exists a fast decaying solution with rate 2θ = n − 2. Now, Namely, (2.2) with the fast rate 2θ = n−2 is a solution for some double bounded c(r).
Step 2. We prove (2.1) has no positive solution when 1 ≤ p ≤ n n−2 . Otherwise, let u be a positive solution. Take x 0 ∈ R n and denote B R (x 0 ) by B. Let Here δ is a Dirac function at x 0 . Then, Here ν is the unit outward normal vector on ∂B. Noting ∂ ν φ < 0, we have According to Proposition 2.1, there exists R j → ∞ (we still denote it by R) such that Thus, noting p ≥ 1, we can use the Hölder inequality to deduce that Write w(x) = c 1/p (x)u(x), then w solves the integral equation involving the Newton potential However, this integral equation has no positive solution for any double bounded c when 0 < p ≤ n n−2 . The proof is a special case of the corresponding proof of Theorem 2.3, which handles a more general integral equation involving the Riesz potential.  Proof. When |x| ≤ 2R for some R > 0, u(x) is proportional to R n |x − y| α−n u p (y)dy. Thus, we only consider the case of |x| > 2R.
Step 1. Inserting (2.2) into the right hand side of (2.5), we can find some double bounded function c(x) such that as |x| > 2R for some R > 0, If p > n n−α , we take 2θ = α p−1 and hence α < 2pθ < n. Then, for some double bounded function c(x). This result shows that (2.5) has the slowly decaying radial solution as (2.2).
Moreover, we can also find a fast decaying solution. Now, take 2θ = n − α, then 2pθ > n as long as p > n n−α . Thus, for some double bounded function c(x).
Step 2. We prove (2.5) has no positive solution when 0 < p < n n−α . Suppose u is a positive solution, then it follows a contradiction. In fact, when Here a 0 = n − α. Using this estimate, for |x| > R we also get By induction, we can obtain where j = 0, 1, · · · , and a j = pa j−1 − α.
In view of 0 < p < n n−α , we claim that {a j } is decreasing. In fact, This induction shows our claim.
Next we claim that there exists j 0 such that Once it is verified, then It is impossible.
Step 3. We prove (2.5) has no positive solution when p = n n−α .
Thus, taking p powers of (2.8) and integrating on B, we have Here c is independent of R. Letting R → ∞, we see u ∈ L p (R n ).
Taking p powers of (2.8) and integrating on Letting R → ∞, and noting u ∈ L p (R n ), we obtain R n u p (y)dy = 0, which contradicts with u > 0.
has a positive solution for some double bounded c(x), if and only if p > n n−2k . Proof. If u > 0 solves the integral equation (2.5) with α = 2k, it is easy to see that u also solves the higher order semilinear PDE (2.10). On the contrary, if p > 1 and u solves (2.10), [37] proved (−∆) i u > 0 for i = 1, 2, · · · , k − 1. Similar to the argument in [11], (2.10) is equivalent to (2.5) with α = 2k. Therefore, if p > 1, Theorem 2.3 shows that (2.10) has positive solutions for some double bounded function c(x), if and only if p > n n−2k .

Integral equation involving the Wolff potential
Theorem 2.5. The Wolff type integral equation has a positive solution for some double bounded c(x), if and only if Proof.
Suppose that u solves (2.11), then (2.14) Next, we consider the case p γ−1 ∈ ( βγ n−βγ , n n−βγ ). Now (2.14) leads to Suppose that a k < a k−1 for k = 1, 2, · · · , j − 1. By virtue of (2.12), it follows Thus, {a j } ∞ j=0 is decreasing as long as (2.12) is true. Furthermore, we claim that there must be j 0 > 0 such that a j0 ≤ 0. This leads to u(x) = ∞, which contradicts with the fact that u is a positive solution.
For R > 0, denote B R (0) by B R . By using the Hölder inequality, from (2.11) we deduce that for any Hence, exchanging the order of the integral variables, we have Therefore, we get Integrating on B R/4 and using p = n(γ−1) n−βγ again, we get Here c is independent of R. Letting R → ∞ and noting p > γ − 1, we have (2.17) where c is independent of R. Letting R → ∞, and noting (2.17), we obtain R n u p (y)dy = 0, which implies u ≡ 0. It is impossible. The proof is complete.

γ-Laplace equation
has positive solutions for some double bounded c(x).
(2) Suppose u solves (2.18) and satisfies inf R n u = 0. According to Corollary 4.13 in [21], there exists C > 0 such that Since c(x) is double bounded, we can see that is also double bounded. This shows that u solves When 0 < p ≤ n(γ−1) n−γ , Theorem 2.5 shows that this Wolff type equation has no positive solution for any double bounded function K(x). Therefore, we prove the nonexistence of positive solutions to (2.18) when 0 < p ≤ n(γ−1) n−γ .
3 Systems with variable coefficients 3.1 HLS type system for some double bounded functions c 1 (x) and c 2 (x), if and only if pq > 1 and Similar to the argument in the proof of Theorem 2.3, we can find four pairs solutions.
(i) Take the slow rates Then pq > 1 as well as (3.2) lead to α < 2pθ 1 < n and α < 2qθ 2 < n. Therefore, for some double bounded functions c 1 (x) and c 2 (x). This consequence shows that (3.1) has a pair of radial solutions (u, v) as (3.3).
By the same argument above, we know that once pq > 1 as well as α n−α < q < n n−α and (p+1)α pq−1 < n − α, (3.1) has a pair of radial solutions (u, v) as (3.3). Now, u, v decay fast by two different rates.
(iv) We can find another pair of radial solutions to (3.1). They decay with fast rates which are different from (3.3). Now, we assume It is easy to verify that u, v also solve (3.1) with some double bounded functions c 1 , c 2 .
These results imply u(x) = ∞ or v(x) = ∞. It is impossible. The contradiction shows the nonexistence of the positive solutions to (3.1).
(ii) If pq > 1 and max{ (p+1)α pq−1 , (q+1)α pq−1 } = n − α, we prove the nonexistence. The idea is the same as Step 3 in the proof of Theorem 2.19. Denote B R (0) by B. First, Without loss of generality, assume p ≤ q. Combining two results above with where c is independent of R. Letting R → ∞, we get v ∈ L q (R n ). On the other hand, we also obtain Letting R → ∞ and noting v ∈ L q (R n ), we see v ≡ 0. It is impossible.
Theorem 3.1 is proved.
Corollary 3.2. Let k ∈ [1, n/2) be an integer and pq > 1. There exist positive solutions u, v of the semilinear Lane-Emden type system for some double bounded functions c 1 (x) and c 2 (x), if and only if Proof. When pq > 1, Liu, Guo and Zhang [37] proved (−∆) i u > 0 and (−∆) i v > 0. Similar to the argument in [8] we can also establish the equivalence between (3.6) and (3.1). So Corollary 3.2 is a direct corollary of Theorem 3.1 with α = 2k.

Wolff type system
for some double bounded functions c 1 (x) and c 2 (x), if and only if pq > (γ − 1) 2 and . Proof.
Step 1. Existence. Insert (3.3) into W β,γ (u p ) and W β,γ (v q ). Similar to the argument in the proof of Theorem 2.5, we also discuss in four cases.
(iii) Similar to the argument in Theorem 3.1, if pq > (γ − 1) 2 , the condition is also stronger than (3.8). When this stronger condition holds, then we take Therefore, βγ < 2pθ 1 < n and 2qθ 2 > n, and hence for another double bounded functions c 1 (x), c 2 (x). This shows (3.7) has radial solutions as (3.3). Similar to the argument above, if another stronger condition pq > (γ − 1) 2 as well as holds, (3.7) also has radial solutions as (3.3) with two different fast rates 2θ 2 = n−βγ also has another pair of radial solutions which also decay fast by two different rates. One decays with n−βγ γ−1 , and another decays with logarithmic order. Now, we assume ; v(x) = (log |x|) .
It is easy to verify that u, v solve (3.7) with some double bounded functions c 1 , c 2 .
Assume u, v are positive solutions of (3.7). Noting BR(0) v q (y)dy ≥ c, we obtain that for |x| > R, Here a 0 = n−βγ γ−1 . By this estimate, for |x| > R, there holds When βγ − pa 0 ≥ 0, we see v(x) = ∞ for |x| > R. This implies the nonexistence of positive solutions of (3.7) since R is an arbitrary positive number.

14)
then there exist positive solutions u, v of the γ-Laplace system for some double bounded c 1 (x) and c 2 (x).
(2) For any double bounded functions c 1 (x) and c 2 (x), if one of the following conditions holds:

Similar to the calculation in (2.19), we also obtain
Therefore, the signs of both sides of the results above show four cases.
(2) Nonexistence. Suppose u, v are positive solutions of (3.15) satisfying inf R n u = inf R n v = 0. According to Corollary 4.13 in [21], there exists C > 0 such that Since c 1 and c 2 are double bounded, we can find two other double bounded functions K 1 (x) and K 2 (x) such that By Theorem 3.3 with β = 1, we can see the nonexistence.

Finite energy solutions: scalar equations
In this section, we consider the critical conditions associated with the existence of the positive solutions when the coefficient c(x) ≡ Constant. Without loss of generality, we take c(x) ≡ 1. For the higher order equation, the corresponding result above is still true. Furthermore, we have the more general result. Proof. Take the scaling transform

Critical exponents and scaling invariants
Then If u µ still solves (4.1), then If the L p+1 (R n )-norm is invariant, then there holds σ = n p + 1 .

Combining this with (4.3), we get (4.2).
On the contrary, if (4.2) is true, then we can also deduce the invariance by the same calculation above.
Proof. Take the scaling transform Then

Thus, u µ solves (4.4) if and only if
Next, Combining this with (4.7), we get (4.5). At last, The L p+1 (R n )-norm is invariant, if and only if σ = n p + 1 .
Since the corresponding result of the γ-Laplace equation can not be covered by that of the Wolff type equation, we should point out the following conclusion. Proof. Suppose u µ is a solution of (4.8). Then Let y = µx, then −µ σ(γ−1)+γ div y (|∇ y u(y)| γ−2 ∇ y u(y)) = µ pσ u p (y).

This result shows that the equation is invariant if and only if
By the same argument as in Theorem 4.2, the invariance of the energy is equivalent to σ = n p + γ − 1 .
Eliminating σ from the two formulas above yields p = n+γ n−γ (γ − 1). The proof that (4.6) with β = 1 is the sufficient and necessary condition is the same as the argument above.
A classical positive solution u ∈ L 2 * (R n ) of (4.9) is called finite energy solution, if u ∈ L p+1 (R n ) or ∇u ∈ L 2 (R n ).
Next, we use the Pohozaev type identity in integral forms to discuss the existence of the finite energy solutions of (4.1). A positive classical solution u of (4.1) is called finite energy solution, if u ∈ L p+1 (R n ). Proof. If (4.2) holds, (4.1) exists a unique class of finite energy solutions (cf. [11] or [35]): Here c, t are positive constants.
On the contrary, if u ∈ L p+1 (R n ) solves (4.1), we claim that (4.2) is true. In fact, for any µ = 0, from (4.1) it follows Differentiate both sides with respect to µ. Then, Letting µ = 1 yields To handle the last term of the right hand side of (4.16), we integrate by parts to get for any R > 0. Here B R = B R (0) and Next, we claim the first term of the right hand side of (4.17) converges to zero as R → ∞. In fact, for suitably large R, (4.18) By Theorem 2.3, we see that p ≥ n n−α . So α − n + n p+1 < 0. In addition, using Proposition 2.1 and u ∈ L p+1 (R n ), we can find R j → ∞ such that Let R = R j → ∞ in (4.18), we verify our claim. Multiplying (4.16) by u p (x) and applying the claim above, we obtain On the other hand, integrating by parts an using (4.19), we get Combining this with the result above, we deduce that This is (4.2). Theorem 4.5 is proved.
Corollary 4.6. Let k ∈ [1, n/2) be an integer and p > 1. The 2k-order Lane- has positive classical solution in L p+1 (R n ) if and only if p = n+2k n−2k . Proof. When p > 1, Corollary 2.4 shows that (4.20) is equivalent to the HLS type equation (4.1) with α = 2k. According to Theorem 4.5, we have the corresponding critical conditions p = n+2k n−2k for the existence of the finite energy solutions of the (4.9). Remark 4.1. Theorem 4.5 shows another critical condition (4.2) for the existence of the positive solutions to (4.1). Since the finite energy solutions class of (4.1) is smaller than the positive solutions class of (2.5), the critical condition (4.2) is stronger than (2.6).
To define the finite energy solution, we first introduce the following theorem. It is a natural generalization of Theorem 4.4.
A classical positive solution u ∈ L γ * (R n ) of (4.8) is called finite energy solution if u ∈ L p+1 (R n ) or ∇u ∈ L γ (R n ).
Proof. Let u ∈ L γ * (R n ). Take a cut-off function ζ R as (4.10). Using the Hölder inequality, we get where D = B 2R (0), and C > 0 is independent of R.
1. For the Wolff type equation (4.4), we do not know whether (4.5) is the necessary and sufficient condition for the existence of positive solution in L p+γ−1 (R n ).

2.
A surprising observation is, when γ = 2, the critical condition (4.26) is different from (4.5) with β = 1. One reason is that the solution of (4.8) only solves a Wolff type equation with variable coefficient instead of (4.4). Another reason is that the finite energy functions spaces L p+1 (R n ) and L p+γ−1 (R n ) are also different except for γ = 2. This distinction shows that (4.2) and (4.26) are not the same class critical exponents. For γ-Laplace equation, besides the divided number in Theorem 2.6, we also have two critical exponents mentioned above. The relation of them is as long as γ ∈ (1, 2). This is also led to by the difference of the existence spaces of positive solutions.

Eq. (5.3) is invariant if and only if
Namely, Energy integrals u L p+γ−1 (R n ) and v L q+γ−1 (R n ) are invariant if and only if Eliminating σ 1 and σ 2 , we obtain (5.4).
By the same calculation in (2), energy integrals u L p+γ−1 (R n ) and v L q+γ−1 (R n ) are invariant if and only if Eliminating σ 1 and σ 2 , we deduce (5.9). By the same argument in (2), (5.10) is another corresponding sufficient and necessary condition.

Existence and the critical conditions
In this subsection, we first show that (5.2) is the critical condition of the existence of the finite energy solution of (5.1). We call the positive classical solutions u, v of (5.1) finite energy solutions, if u ∈ L p+1 (R n ) ∩ L 2 * (R n ), and v ∈ L q+1 (R n ) ∩ L 2 * (R n ). Proof. Serrin and Zou [46] proved the existence if (5.2) is true. Next, we will deduce (5.2) from the existence. Denote B R (0) by B. According to Proposition 5.1 in [45] (or cf. Lemma 2.6 in [48]), the solutions u, v satisfy the Pohozaev type identity where a 2 , a 2 ∈ R satisfy a 1 + a 2 = n − 2. Since u, v are finite energy solutions, we know ∇u, ∇v ∈ L 2 (R n ) by an analogous argument of Theorem 4.4. Using Proposition 2.1 and the Young inequality, we can find R j → ∞, such that all the terms in the right hand side converge to zero. Letting R = R j → ∞ in the Pohozaev identity above, we obtain for any a 1 , a 2 as long as a 1 + a 2 = n − 2. Take a 2 = n q+1 , then This implies 0 = n p+1 − a 1 = n p+1 − (n − 2 − a 2 ) = n p+1 − (n − 2) + n q+1 . So (5.2) is verified.
Next, we consider the HLS type system. Since (5.6) is the Euler-Lagrange system of the extremal functions of the HLS inequality which implies (u, v) ∈ L p+1 (R n )×L q+1 (R n ), we naturally call such solutions (belonging to L p+1 (R n )× L q+1 (R n )) of (5.6) as finite energy solutions. Proof. Sufficiency. Clearly, the extremal functions of the HLS inequality are the finite energy solutions. Lieb [35] obtained the existence of those extremal functions.
Necessity. The Pohozaev type identity in integral forms is used here. For any µ = 0, there holds Differentiate both sides with respect to µ and let µ = 1. Then, (5.14) According to Remark 1.2 (1) (or cf. Theorem 1 in [6]), if p, q ≤ α n−α , (5.6) has no any positive solution. Therefore, (u, v) solves (5.6) implies p, q > α n−α . Similar to the derivation of (4.18), if follows when R = R j → ∞. Thus, integrating by parts, we obtain Multiplying (5.14) by v q (x) we get Similarly, there also holds On the other hand, integrating by parts leads to and similarly R n u p (x)(x · ∇u(x))dx = −n p+1 R n u p+1 (x)dx. Inserting these into the result above, we deduce that From (5.6), it follows that Substituting this into the result above yields Theorem 5.3 is proved.
In fact, This proves that for a < ε 0 , we can find R ∈ (0, 1] such that u a (r), v a (r) > 0 for r ∈ (0, R) and v a (R) = 0.
Clearly, S = ∅ by virtue of ε 0 ∈ S. Noting ε ≤ a 0 for ε ∈ S, we see the existence of a.
Otherwise, there existsR > 0 such thatū(r),v(r) > 0 for r ∈ (0,R) and one of the following consequences holds: We deduce the contradictions from three consequences above. (i) By C 1 -continuous dependence of u a , v a in a, and the factū ′ (R) < 0, we see that for all |a − a| small, there exists R a > 0 such that u(r),v(r) > 0, f or r ∈ (0, R a ); u(R a ) = 0,v(R a ) > 0.
This contradicts with the definition of a.

Nonexistence in bounded domain
In this subsection, we give the Pohozaev identity which the proof of Theorem 6.1 needs. In fact, we can give more general ones which imply nonexistence of positive solutions of the following 2k-order PDE (1.6) (−∆) k u = u p , k ≥ 1, u > 0 with the supercritical exponent p > n+2k n−2k in any bounded domain. The argument can help to prove the existence results in R n .
Note. Seeing here, we recall another related fact: in the subcritical case, (1.6) has positive solutions in a bounded domain. In general, the variational methods works now. However, it has no positive solution in R n (cf. Remark 6.1(3)).
By the maximum principle, from −∆u k = u p 1 > 0 and u k | ∂D = 0, we see u k > 0 in D. By the same way, we also deduce by induction that u j > 0 in D, j = 1, 2, · · · , k. (6.7) Multiplying the j-th equation by (x · ∇u k+1−j ), we have for j = 1, 2, · · · , k, where ν is the unit outward normal vector on ∂D.