LIOUVILLE THEOREMS AND CLASSIFICATION RESULTS FOR A NONLOCAL SCHR¨ODINGER EQUATION

. In this paper, we study the existence and the nonexistence of positive classical solutions of the static Hartree-Poisson equation where n ≥ 3 and p ≥ 1. The exponents of the Serrin type, the Sobolev type and the Joseph-Lundgren type play the critical roles as in the study of the Lane-Emden equation. First, we prove that the equation has no positive solution when 1 ≤ p < n +2 n − 2 by means of the method of moving planes to the following system When p = n +2 n − 2 , all the positive solutions can be classiﬁed as with the help of an integral system involving the Newton potential, where c,t are positive constants, and x ∗ ∈ R n . In addition, we also give other equivalent conditions to classify those positive solutions. When p > n +2 n − 2 , by the shooting method and the Pohozaev identity, we ﬁnd radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate 2 p − 1 . Finally, we point out that the equation has positive stable solutions if and only if p ≥ 1 + 4 n − 4 − 2 √ n − 1 .

with the help of an integral system involving the Newton potential, where c, t are positive constants, and x * ∈ R n . In addition, we also give other equivalent conditions to classify those positive solutions. When p > n+2 n−2 , by the shooting method and the Pohozaev identity, we find radial solutions for the system. In particular, the equation has a radial solution decaying with slow rate 2 p−1 . Finally, we point out that the equation has positive stable solutions if and only if p ≥ 1 + 1. Introduction. This paper is concerned with the static Hartree-Poisson equation where n ≥ 3, and p ≥ 1.
Such an equation arises in the Hartree-Fock theory of the nonlinear Schrödinger equations (cf. [33]). Equation (1.1) is also helpful in understanding the blow-up or the global existence and scattering of the solutions of the dynamic Hartree equation in the focusing case (cf. [26]). A more general form is the Choquard type equation studied in many papers (such as [14], [22], [31], and [38]), which arises in the study of boson stars and other physical phenomena, and also appears as a continuous-limit model for mesoscopic molecular structures in chemistry. More related mathematical and physical background can be found in [15], [34], [40] and the references therein. The nonlocal term in (1.1) appears in the example 3.2.8 of the book [4], and it is also related to a simplified model of the Schrödinger-Poisson system (cf. [1], [18], [19] and many others).
The existence of the super-solutions of (1.1) was studied in [39] and several sufficient conditions were listed. However, it seems difficult to investigate directly the existence of positive solutions in view of the convolution term. Write v(x) = √ p R n u p (y)dy |x − y| n−2 .
Then v > 0 in R n . Noting the relation between the Newton potential and the convolution properties of Dirac function, we see that there holds formally − ∆v(x) = √ p(−∆|x| 2−n ) * u p = √ pδ x * u p = √ pu p (x), (1.2) where δ x is the Dirac mass at x. Thus, the positive solutions of (1.1) must satisfy the following system Quittner and Souplet [41] studied positive solutions of more general PDE system They proved the following results: (Rt1) If n ≥ 3, p − s = q − r ≥ 0 and 0 ≤ r, s ≤ n n−2 , then positive solutions u, v of (1.4) satisfy u ≡ v. It is called the symmetry of components in [41].
In addition, Ma and Liu proved the radial symmetry for the decay solutions of (1.4) by the method of moving planes (cf. [37]).
According to (Rt1), one gets from (1.3) that u ≡ v when 1 ≤ p ≤ 2n−2 n−2 . Now, (1.3) is reduced to the Lane-Emden type equation − ∆u = √ pu p , u > 0 in R n . (1.5) The classification of the solutions of this Lane-Emden equation (1.5) has provided an important ingredient in the study of conformal geometry, such as the prescribing scalar curvature problem and the extremal functions of the Sobolev inequalities. It was studied rather extensively. Recall existence results of this single equation. By the Liouville type result in [13], (1.5) has no positive classical solution when 1 ≤ p < n+2 n−2 . When p = n+2 n−2 , Gidas, Ni, and Nirenberg [12] pointed out that all the classical solutions of (1.5) with a reasonable behavior at infinity must be of the form where the constants c, t > 0, and x * ∈ R n . Later, Caffarelli, Gidas, and Spruck [2] removed the decay restriction and obtained the same result. Then Chen and Li simplified their proof (cf. [5] [24]). The method of moving planes comes into play in those work. In addition, the method of moving sphere introduced by Li and Zhu [29] is also an important approach for researching (1.5) (see also [28]).
In this paper, we are concerned with the Liouville type theorems and the classification results on positive solutions of (1.1) via studying (1.3). Our motivation is to answer whether those existence/nonexistence results are analogous to the corresponding conclusions of the Lane-Emden equation. The main challenge is how to handle the incompletely coupling terms of (1.3) (comes from the nonlocal term of (1.1)). The following integral system plays an important role in studying (1.3) with some constants c 1 , c 2 > 0 (cf. [7] and [23]).
Remark 1.1. The integral system (1.7) is much close to the classical integral equation of the Hardy-Littlewood-Sobolev type (cf. [7], [27] and [32]) and hence its properties are predictable. In addition, the referee pointed out that a more general integral system corresponding to the PDE system (1.4) is also an interesting model. However, differential from the result in [41], the symmetry of components of the integral system seems difficult to be obtained, and hence it cannot be reduced to a single equation. Although this integral system is more complicated, the properties are obviously more abundant and we would study it in the future. In general, p is called a subcritical exponent, critical exponent, and supercritical exponent, if p < 2 * − 1, p = 2 * − 1, and p > 2 * − 1 respectively, where 2 * = 2n n−2 . We state the main results in three cases.
1.1. Subcritical case p ∈ (1, 2 * − 1). When n ≥ 4, (1.3) is reduced to (1.5) by (Rt1). So the Liouville theorem of the Lane-Emden equation is already known in subcritical case (cf. [13]). Consider the case of n = 3. In view of the convolution term, the standard profile in [13] does not work in studying the nonexistence result. In Section 2, we apply the method of moving planes employed by Chen-Li [5], Ma-Chen [36] and Ma-Liu [37] to study (1.3) and prove the following theorem.
The nonexistence for (1.7) is also considered in Section 2. It indicates that the Liouville type result may be true for other weaker positive solutions of (1.1) (e.g. for some integrable solutions).
Remark 1.2. The Serrin exponent n n−2 is a well known exponent because not only it appears in the trace embedding inequality (cf. [11]), but also it is critical for the existence of super-solutions of (1.5). In addition, the Serrin exponent is also critical for the existence of positive solutions of the Lane-Emden equation with 3) implies that w satisfies the equation above. However K(x) only satisfies a weaker condition 0 < K(x) ≤ C. Now, both the nonexistence in the case of 1 ≤ p ≤ n n−2 and the existence in the case of p > n n−2 are unclear. So the Serrin exponent seems difficult to be used to study (1.3)  (R n )-solutions are important. In fact, Li and Ma [25] proved that the positive solutions in L n(p−1) 2 (R n ) are radially symmetric and decreasing. Jin and Li [20] applied a regularity lifting lemma by the contraction maps to obtain the optimal integrability of positive integrable solutions. Based on this result, [42] estimated the fast decay rates when |x| → ∞. All the results show that those integrable solutions (i.e. L n(p−1) 2 (R n )-solutions) of (1.7) have better regular properties (see also [27]). So we keep the focus on this class of solutions in the following argument.
1.2. Critical case p = 2 * − 1. When n ≥ 4, one can classify easily the classical solutions by (Rt1) and the results in [5]. In Section 3, we not only discuss the classification results in the case of n = 3, but also give other conditions for classifying the positive solutions. Here, the integral system (1.7) comes into play. Remark 1.4. Another integral system related to (1.7) is the following Lane-Emden type equations involving the Riesz potentials It is essential in studying the extremal functions of the Hardy-Littlewood-Sobolev inequality (cf. [32]) |x − y| λ dxdy ≤ C(n, s, λ)||f || r ||g|| s with 0 < λ < n, 1 < s, r < ∞, f ∈ L r (R n ) and g ∈ L s (R n ), 1 r + 1 s + λ n = 2. Define T g(x) = R n g(y) |x − y| n−α dy with α = n − λ. The Hardy-Littlewood-Sobolev inequality becomes ||T g|| p ≤ C(n, s, α)||g|| np n+αp . (1.10) This inequality will be used in this paper to research the radial symmetry and the integrability of the solutions of (1.7).
In the critical case, classification results for the single equation of (1.9) can be found in [7] and [27]. We here study the integral system (1.7) and then obtain several necessary and sufficient conditions for the classification results of (1.3).
Theorem 1.4. Let (u, v) be a pair of classical solutions of (1.3) with p > 1. Then the following items are equivalent to one another (1) u ∈ L n(p−1) 2 (R n ); (2) u is bounded and decays with the fast rate n − 2; (3) u belongs to the homogeneous Sobolev space D 1,2 (R n ); where u * is the radial function as the form (1.6).
1.3. Supercritical case p > 2 * − 1. Finally, we consider the supercritical case p > 2 * − 1 in Section 4. The existence and nonexistence are much complicated and not completely understood even for the Lane-Emden equations.
First (1.1) has a singular radial solution in R n \ {0} p−1 (which is a direct corollary of Theorem 4.5), as long as p is larger than the Serrin exponent n n−2 . Next, we can find bounded entire solutions. An example is bounded radial solutions. Theorem 1.6. When p > 2 * − 1, we can find radial solutions of (1.3). In addition, there is a radial solution of (1.1) decaying with the slow rate 2 p−1 when |x| → ∞. In fact, we can use the shooting method introduced in [35] to find a solution of the following system of ODEs (1.12) Interestingly, it seems difficult to show that V decays to zero when r → ∞. On the other hand, for the system we can see that V goes to zero when r → ∞ if noting the integral form of V .
Remark 1.5. Not all the solutions in the supercritical case are radially symmetric. There is another example to show that some bounded entire solutions are neither radial nor decaying when |x| → ∞. Introduce a pair of cylinder-shaped solution (u * , v * ) to (1.3) (cf. [8]). According to Theorem 1.4, (u * , u * ) solves (1.3) in the whole space R n in the critical case p = n+2 n−2 , where u * is of the form (1.6). Thus, it is not difficult to see that u * (x, x n+1 ) = u * (x) and v * (x 0 , x) = u * (x) still solve (1.3) in R n+1 . In view of p > n+3 n−1 , the problem (1.3) (which u * , v * satisfy in R n+1 ) is equipped with a supercritical exponent. Clearly, this pair of positive solution (u * , v * ) is neither radial nor decaying when |x| → ∞. We also see u * = v * because the generating lines of the cylinders are different.
We also consider the nonexistence of 'stable' positive solutions of (1.1). The Joseph-Lundgren type exponent comes into play (cf. [16]), which is also essential to describe how the radial solutions intersect with the singular radial solution (1.11) and with themselves (cf. [21]). In addition, this Joseph-Lundgren exponent can be used to study the Morse index for the sign-changed solutions of the Lane-Emden equation (cf. [10]) and other semilinear elliptic equations with supercritical exponent (cf. [17]). (1.14) Here This definition is well-defined. Indeed, (1.14) comes from the fact that the ei- has a first positive eigenvalue η > 0 with corresponding positive eigenfunction φ.
On the other hand, although (1.14) cannot be deduced directly from the positive definite quadratic form of the functional some comparison relation of u and v (similar to Lemma 2.4) shows that the right hand side of (1.14) makes sense. More importantly, for such a definition of stable solutions, we can prove that the exponent p jl (n) is critical on existence and nonexistence.
(1.1) has no positive stable solution.

2.1.
Liouville theorem for integral system. Proof. When p = 1, if we write w = u + v, then (1.7) is reduced to According to Theorem 1.4 in [27], we see the conclusion. When 1 < p ≤ n n−2 . If u, v are positive solutions of (1.7), we can deduce a contradiction by following the ideas in [3].
By (1.7), we have In addition, for any R > 2, from where c > 0 is independent of R.
When 1 < p < n n−2 , we can see a contradiction by letting R → ∞ in (2.2).
we also deduce Using (2.1), (2.3) and noting p = n n−2 , we get By using the method of moving planes in integral form, which was established by Chen-Li-Ou (cf. [7] and [8]), we prove a radial symmetry result. (2.5)

YUTIAN LEI
Then the positive continuous solutions of are radially symmetric and decreasing around x * ∈ R n . Moreover, x * = 0 as long as h > 0.
Since the second term of the right hand side is nonpositive, from the definition of Σ u λ and Σ v λ , it follows that Using the Hardy-Littlewood-Sobolev inequality (1.10) and the Hölder inequality, we have Combining these results, we can see that Σ u λ and Σ v λ are empty set as long as λ is near −∞.
Suppose that at . By the same argument as above, we can prove that there exists an > 0, such that u(x) ≥ u λ (x) and v(x) ≥ v λ (x) on Σ λ for all λ ∈ [λ 0 , λ 0 + ). Therefore, we can move the plane x 1 = λ to the right as long as hold on Σ λ . If the plane stops at x 1 = λ 0 for some λ 0 < 0, then u(x) and v(x) must be radially symmetric and decreasing about the plane x 1 = λ 0 . Otherwise, we can move the plane all the way to x 1 = 0. Since the direction of x 1 can be chosen arbitrarily, we obtain that u(x), v(x) are radially symmetric and decreasing about some x * ∈ R n .
If h = 0, we claim x * = 0. Otherwise, we can find λ 0 < 0 such that x 1 = λ 0 is the plane at which we have to stop. From (2.7), we get (1.7). According to Theorem 2.2 with h = 0, we see that they are radially symmetric about x * ∈ R n . Since (1.7) is invariant after translation, x * can be chosen arbitrarily.
Step 2. Consider the Kelvin transformation of u, v (2.8) By (1.7), we see thatū,v solve (2.6) with h = n + 2 − p(n − 2). In view of the fact that p < 2 * − 1, it follows h > 0. In addition, from u, v ∈ L n(p−1) 2 (R n ), we see that (2.5) forū,v is true. According to Theorem 2.2,ū,v are also radially symmetric but the center point x * must be the origin. So the translation invariant is absent. By the same argument of Theorem 3 in [8], we can also deduce a contradiction.

2.2.
Liouville theorem for PDEs. First, we have a comparison result as (Rt2) and the ideas in [41] are used in the proof.
Integrating from 0 to R > 0 yields v A ≤ −cR. Integrating again and letting R sufficiently large, we see that v A is negative. It is impossible. Thus, there exists r j → ∞ such that (H 2 ) A (r j ) ≤ 0. Inserting this into (2.10) with R = r j implies H ≡ C and hence H A ≡ C. Combining with (2.11) yields C = 0. Namely, H ≡ 0 and then u ≤ v on R n . Lemma 2.4 implies that the solution u of (1.3) is also a super-solution of (1.5), i.e. −∆u ≥ u p on R n . Thus, if p is not larger than the Serrin exponent n n−2 , (1.3) has no positive classical solution. Moreover, we have the following stronger result.
n−2 , then there does not exist any positive classical solution of (1.3).
Proof. Here we use the method of moving planes to prove that the positive classical solutions u, v are radially symmetric which implies the nonexistence.
Since u, v are classical solutions, u(0) and v(0) are finite. Therefore, from (2.8) it follows that the Kelvin transformations of u, v satisfȳ u,v |x| 2−n when |x| → ∞. (2.12) In addition,ū andv are the positive solutions of (2.13) Once (2.13) is true, we can start to move plane {x 1 = λ} from −∞ to the origin.
If W 2 (x 0 ) < 0, we can assume that W 2 reaches its negative minimum at y 0 . By the same derivation of (2.15), we also have with positive constants c 4 , c 5 . Combining with (2.15) yields . In view of W 1 (x 0 ) < 0, the result above is impossible for large |x 0 ||y 0 |. This contradiction shows U λ > 0. Similarly, we can also deduce V λ > 0, and hence (2.13) is proved.
By an analogous argument above, we can also use the comparing principle to establish that there is R 0 > 0 which is independent of λ such that if x 0 , y 0 are the negative minimal value points of W 1 , W 2 respectively, then |x 0 |, |y 0 | ≤ R 0 . This conclusion ensures that the plane can move to its right limit ∂Σ λ0 . Here The rest proof is standard by means of the method of moving planes (cf. [5] or [6]) andū,v are radially symmetric and decreasing about the origin. Therefore, u, v are also radially symmetric and decreasing. Since the (1.3) is invariant under translation, the symmetry point can be arbitrary. It also leads to a contradiction and the nonexistence is obtained.
Remark 2.1. It should be pointed out that the elliptic methods in [13] still work if u, v are radially symmetric. In fact, when n ≥ 4, it follows p − 1 ≤ n n−2 . Now, we can apply Theorem 1.2 in [41] to get u ≡ v and hence (1.3) becomes (1.5). According to Theorem 1.1 in [13], we see the nonexistence. So we consider the case of n = 3 only. When p = 1, if we write w = u + v, then (1.3) is reduced to (1.5). According to Theorem 1.1 in [13], we also see the nonexistence. When p ∈ (1, 2], the elliptic method in Section 2 in [13] is still valid. However, when p ∈ (2, 5), it is difficult to deduce the same conclusion from Lemma 2.4 only, because the partial derivatives of u, v need to be compared. As long as u, v have the radial structure, we can also obtain the estimates by the same argument in [13].
Thus, for sufficiently large |x|, In addition, for sufficiently large |x| we also get easily that Combining the three estimates above, we obtain (3.4).
Letting |x| → ∞ in Similarly, we can also deduce are positive solutions of (1.3) in L n(p−1)/2 (R n ). However, it is nontrivial to show that all solutions of (1.3) in L n(p−1)/2 (R n ) are the above form. In this section, we prove this conclusion.  From (1.7), it follows that w satisfies where 0 < K(x) ≤ C. Set w A = w as |x| > A or w > A; w = 0 as |x| ≤ A and Therefore, w solves the operator equation By the Hardy-Littlewood-Sobolev inequality (1.10), we get Take A suitably large such that C w A p−1 n(p−1) 2 < 1. Thus, T is a contraction map from L s (R n ) to itself. In view of n(p−1) 2 > n n−2 (which is implied by (3.5)), T is also a contraction map from L n(p−1)/2 (R n ) to itself. By the lifting lemma on the regularity (cf. Theorem 3.3.1 in [6] or Lemma 2.1 in [20]), we obtain w ∈ L s (R n ) for s > n n−2 . Thus, u, v ∈ L s (R n ) for s > n n−2 . On the other hand, if s ≤ n n−2 , by (2.1) we have Similarly, we also deduce v ∈ L s (R n ) for all s ≤ n n−2 . (R1) is proved.
Clearly, for a suitably small > 0, from (R1) we deduce On the other hand, in virtue of (3.5) and (R1), we get w ∈ L p (R n ). Thus K 2 ≤ Cd 2−n w p p ≤ C. Combining the estimates of K 1 and K 2 , we know that w is bounded. Thus, u, v are bounded.
Next, we show that w is decaying. Take x 0 ∈ R n . By exchanging the order of the integral variables, we have
Step 2. Take a smooth function ζ(x) satisfying Define the cut-off function Multiplying the first equation of (1.3) by uζ 2 R and integrating on D : Integrating by parts, we obtain Applying the Cauchy inequality, we get for any δ ∈ (0, 1/2). According to Theorem 3.2 (R1), it follows that u, v ∈ L 2 * (R n ). By the Hölder inequality, we obtain (3.14) Noting that u p v ∈ L 1 (R n ), from (3.12)-(3.14) we deduce D |∇u| 2 ζ 2 R dx ≤ C. Letting R → ∞ yields R n |∇u| 2 dx < ∞. Similarly, we also obtain R n |∇v| 2 dx < ∞. Combining the results above, we can see that Therefore, we can find R j such that Step 3. Multiplying the first equation of (1.3) by u and integrating on D, we get Here ν is the outward unit normal vector on ∂D. By the Hölder inequality, from (3.15) we deduce Similarly, we can also obtain ∇v 2 2 = √ p u p v 1 .
The following result shows that there does not exist solution in class L n(p−1) 2 (R n ) if p is not equal to the critical exponent 2 * − 1. (R n ), then p = 2 * − 1, and hence L n(p−1)/2 (R n ) = L 2 * (R n ).
Proof. Write B = B R (0). Multiply two equations of (1.3) by x · ∇u and x · ∇v, respectively. Integrating on B, we get Integrating by parts yields

YUTIAN LEI
Adding the two results together and integrating by parts again, we obtain Letting R = R j → ∞ and using (3.15), we have By (3.10) we see p = 2 * − 1 finally.
According to Theorem 3.1 and Theorem 2.2 with h = 0, the positive classical solutions of (1.3) are radially symmetric and decreasing about x * ∈ R n as long as u ∈ L n(p−1) 2 (R n ). Moreover, we have the following stronger result.
with some constant c = c(n) and for some t > 0. Proof.
Step 1. We claim u ≡ v.
Let W = u − v. By Theorems 3.1-3.3, we see that R n (|W | 2 * + |∇W | 2 )dx < ∞. Thus, when R = R j → ∞, Here B = B R (0). By the Hölder inequality and (3.18), as R = R j → ∞, Step 2. By virtue of u ≡ v and Theorem 3.4, (1.3) is reduced to the single equation According to the classification results in [5], the positive solutions must be of the form (3.17) in the critical case.
The argument above implies that a classical solution u ∈ L n(p−1) 2 (R n ) is equivalent to (3.17).
At last, we complete the proof of Theorem 1.4.

Radial solutions.
In order to find the existence of entire solutions in R n , we need the following nonexistence result on a bounded domain, which is deduced by the Pohozaev identity. then the following boundary value problem has no nontrivial nonnegative radial so-  Since u has radial structure, |∇u| 2 = |∂ ν u| 2 on ∂D. Multiplying (4.2) by (x · ∇u) and integrating on D, we get Integrating by parts and noting (4.4), we obtain Similarly, from (4.3) we also deduce that Combining two results above with (4.5) yields If u, v are nontrivial, then which contradicts with (4.1).
Based on the above Liouville type result, we can search for positive solutions of (1.3) with radial structure. Let u, v be radially symmetric about x * ∈ R n . We can write Theorem 4.2. Let p > 2 * − 1. Then the following ODE system has entire solutions for constant a > 0.
Proof. Here we use the shooting method.
Step 1. First, we know that U and V are not increasing since the right hand sides of equations in (4.6) are positive. By a standard contraction argument, we can see the local existence. We denote the solutions by u a (r), v a (r).
Step 2. We claim that either (4.6) has entire solutions for all a > 1, or for some a * > 1, there exists R ∈ (0, 1] such that u a * (r), v a * (r) > 0 for r ∈ [0, R) and u a * (R) = 0. In fact, integrating (4.6) twice yields Let a > 1. So we can find δ > 0 such that v a (r) > u a (r) for r ∈ [0, δ) by the continuity of u a , v a . We claim v a (r) > u a (r) for all r ≥ 0. Otherwise, there exists r 0 ≥ δ such that v(r 0 ) = u(r 0 ) and v(r) > u(r) as r ∈ [0, r 0 ). From (4.7) with r = r 0 we can deduce a contradiction easily. Therefore, if u a (r) > 0 for all r ≥ 0, then (4.6) has entire solutions and the proof is complete. Otherwise, we can find R > 0 such that u a (r), v a (r) > 0 for r ∈ (0, R) and u a (R) = 0. We denote the a in this case by a * .
Step 4. Let a = sup S, where Clearly, S = ∅ in virtue of ε 0 ∈ S. From Step 2, it follows ε ≤ a * for ε ∈ S. Namely, S is upper bounded, and hence 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.
(1) By the 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 );ū(R a ) = 0,v(R a ) > 0.
This contradicts the definition of a.
(2) Similarly, for |a − a| small, there exists R a > 0 such that This implies that a + δ ∈ S for some δ > 0, which contradicts with the definition of a.
(3) The consequence implies that u(x) =ū(|x|) and v(x) =v(|x|) are solutions of the system This is impossible by Theorem 4.1.
All the contradiction arguments show that our claim is true. Thus, the entire positive solutions exist.
In supercritical case, whether the solution (u, v) of (1.3) satisfy symmetry of components u ≡ v is not clear (even (1.3) has radial structure). However, if u is a classical solution of (1.1) and v is the Newton potential of u p , the following theorem shows that the symmetry of components u ≡ v may be true, and hence Theorem 1.6 is a direct corollary.  Proof.
Furthermore, for V (|x|) = √ p|x| 2−n * U p (|x|), from (1.2) it is not difficult to see and hence V (r) < 0 as r > 0. Integrating (4.9) and noting the monotonicity of U and V , we have Eqs. (4.9) and (4.11) imply that u, v also solve (1.3). According to Lemma 2.4, the entire solution u of (1.1) is not larger than v. Therefore, the result above implies U p (r)r 2 ≤ 8n 3 √ p U (r/2). Set G(r) = U (r)r 2 p−1 , then we get an iteration result for any k = 2, 3, · · · . Letting k → ∞, there holds Namely, there exists R > 0 such that for |x| > R, This shows the claim.
Step 3. We claim u ≡ v. The argument in Step 1 of the proof of Theorem 3.5 does not work, since the boundary integral is difficult to handle. Thanks to the work of [25]. Those ideas are powerful to deal with symmetry of components no matter the value of p is critical or not.
Step 4. In virtue of u ≡ v, (1.3) is reduced to the single equation −∆u = √ pu p , u > 0 in R n with (4.1). According to Theorem 2.41 in [30], we know that the radial solution u either decays fast or decays slowly when |x| → ∞. Here c 1 , c 2 are positive constants. If u decays fast, by Theorem 1.4 we know p = 2 * − 1 and u ≡ u * . Here u * is the radial function in (1.6).
If u decays slowly, we claim p > 2 * − 1. Otherwise, from (4.1) we have p = 2 * − 1. According to Theorem 1.4, u ≡ u * , which contradicts with the slow decay rate. Proof. Assume u is a positive stable solution, we can deduce a contradiction.
The following theorem shows that (1.3) has a radial singular solution, which also implies that (1.1) has a singular solution as the form (1.11).