EXISTENCE OF SOLUTIONS TO THE SUPERCRITICAL HARDY-LITTLEWOOD-SOBOLEV SYSTEM WITH FRACTIONAL LAPLACIANS

. It is known that the supercritical Hardy-Littlewood-Sobolev (HLS) systems with an integer power of Laplacian admit classic solutions. In this paper, we prove that the supercritical HLS systems with fractional Laplacians ( − ∆) s , s ∈ (0 , 1), also admit classic solutions.


1.
Introduction. For a bounded C 2 function f : R n → R, fractional Laplacian (−∆) s can be defined by, where (P.V.) stands for the principal value, s ∈ (0, 1) and C n,s is some normalization constant. Equivalently, (P.V.) can be removed and (−∆) s can be defined as In general, fractional Laplacians can be defined (by duality) in a weighted L 1 -space (see [22,34]): L s := f : R n → R such that R n |f (x)| 1 + |x| n+2s dx < ∞ .
The fractional Laplacian operator arises from various scientific backgrounds, e.g., phase transitions, flame propagation, stratified materials and etc. In particular, it has an equivalent definition as an infinitesimal generator of a stable Lévy-process. For a expository reference of the fractional Laplacian, we refer the reader to [35].
In this paper we shall study the following system of equations with fractional Laplacians where n ≥ 2 and p, q satisfy p, q > 1 and 1 p + 1 For 0 < p, q < ∞ and s ∈ (0, n 2 ), (3) is called Hardy-Littlewood-Sobolev (HLS) type system (c.f. [10]) because it has a close connection with the well-known HLS inequality. For the HLS inequality, p, q satisfy the critical condition, 1 p+1 + 1 q+1 = n−2s n . Hence, we call (4) the supercritical condition. The Euler-Lagrange equation of the extremal functions of the HLS inequality, after a change of variables, becomes (3). For the existence of extremal functions, see Lieb [18].
There is also a strong connection between the HLS system and theory of geometric analysis and functional analysis. For example, suppose u = v and p = q, (3) reduces to a scalar equation, (−∆) s u = u p , and there is a large amount of literature about the classification of solutions to this equation, see [2,4,6,13,15] and the reference therein.
For a general HLS type system, 0 < p, q < ∞ and s ∈ (0, n 2 ), the existence of solutions to the supercritical system (3)-(4) is a hard problem, and only some special cases have been settled. For instance, when s = 1, the existence of solutions for (3) can be proved by a shooting method of Serrin and Zou [30]. Later, for an integer s ≥ 2, it is proved by the shooting method with a degree argument (see, c.f. [16,17,19]). However, for a fractional s, less is known. For the single equation, i.e., (3) with u = v and p = q, Chen-Li-Ou [6] have classified the critical case, p = q = n−2s n+2s . For s = 1 2 , the existence of the supercritical single equation, (3)-(4) with u = v and p = q, is obtained by Chipot,et. al. [11], where the authors actually treated an elliptic problem with Neumann boundary condition in R n+1 + . According to the extension method developed by Caffarelli and Silvestre [3], the problem solved by Chipot is equivalent to a Dirichlet problem with (−∆) 1 2 on R n . The main result of this paper is which is radially symmetric and decreases to zero at infinity, and moreover, u(0) + v(0) = 1.
There are difficulties in dealing with a pseudo-differential operator such as the fractional Laplacian (−∆) s due to the nonlocal feature. As a result, many common tools in classical elliptic problems with local operators are not at our disposal. For example, when considering radial equation, the shooting method and ODE theory are very handy for (3) with an integer s, but they are not applicable for a fractional s. To overcome this, our main idea is to use a topological approach, namely the bifurcation theory, to obtain a solution. First, we apply the bifurcation theory to a local auxiliary problem to obtain a solution, and then prove a sequence of solutions to the auxiliary problems converges to a solution of the original problem. Again, due to the lack of locality, difficulties arise in deriving the monotonicity of the solution. We shall implement a direct method of moving plane recently developed by Chen et. al [5] to overcome this issue. For the recent development of the method, see [7,20,36] The paper is organized as follows: In section 2, we prove the existence and the monotonicity of a solution to a local auxiliary problem; in section 3, we show that a sequence of solutions to the auxiliary problems converges to a classic solution to the original problem; in section 4, we use energy estimates to show that the solution decays to zero at infinity with certain rates.
2. Auxiliary problem. Heuristically, we first obtain a non-negative solution (u, v) in R n to the following Dirichlet problem, where B R is a ball in R n with radius R and centered at origin. Then along some sequence R j → ∞ we obtain a sequence of solutions converging to a classic solution of (3).
Remark 1. For a Dirichlet problem involving fractional Laplacians, the equation holds only inside the domain Ω. In fact (−∆) s u is undefined on the boundary ∂Ω when the solution u is only C 0,s (R n ).
To be more precise, if u ∈ C 0,s (R n ) is a solution of the Dirichlet problem above and g ∈ L ∞ (Ω), then for x ∈ R n \ Ω and close to the boundary ∂Ω, by direct Similarly, |(−∆) s u(x)| ≤ C|x| −n−2s for large |x| and |(−∆) s u(x)| ≤ Cδ −s , δ = dist(x, ∂Ω) for x ∈ R n \ Ω and close to the boundary ∂Ω. So (−∆) s u ∈ L p (R n ) with p ∈ [1, 1 s ). We will look for radial solutions of (5) and define a solution space. Let Let G R (x, y) be the Green's function of (−∆) s on B R , which can be obtained by scaling G 1 (x, y), the Green's function of (−∆) s on B 1 (see [26]), with For x ∈ R n \ B R , simply let Af (x), T 1 (f )(x), T 2 (f )(x) be 0.
Obviously, the images of A, T 1 , T 2 are radial, and A, T 1 , T 2 are well defined. Indeed, we will show that they are compact operators on E (see Corollary 1). Now we rewrite (5) into the following system of integral equations, Systems (5) and (11) are closely related, and in suitable sense they are equivalent. Here, we only need the fact that a solution (u, v) ∈ E R to (11) is also a solution to (5).
Let λ 1 be the first eigenvalue and φ 1 the first eigenfunction of the problem and where λ 1 is simple and φ 1 is non-negative (cf. Proposition 9 of [32] by Servadei and Valdinoci). Hence λ 1 is a simple characteristic value of A, and we have Theorem 2.2 directly follows from the following theorem.
In section 2.1, we show that A, T are compact operators in E R . In section 2.2, we prove the first part of Theorem 2.3, namely the existence of solutions, by the Rabinowitz Theorem in the bifurcation theory. Notice that with p, q > 1, we have T 1 (u) = o( u ) and T 2 (v) = o( v ) near (0, 0) in E R , and thus the Rabinowitz Theorem is applicable to (11). In section 2.3, we prove the second part of Theorem 2.3, the monotonicity of the solution, via the method of moving plane.
For simplicity, we denote the operator on E R which maps (u, v) → (Au, Av) still as A, and denote T (u, v) := (T 2 (v), T 1 (u)). We prove that A, T are bounded compact operators on E R in the next section (see estimate (15)). Hereinafter we use the following notation for the Hölder norm in domain Ω ⊂ R n , for s ∈ (0, 1), 2.1. Compactness. Due to a recent study of the regularity of solutions to Dirichlet problems with fractional Laplacians by Ros-Oton and Serra [27], the compactness of A on E directly follows. We shall also give a simple direct proof for the completeness.
Theorem 2.4. The operator A defined in (9) is a compact operator on E, which is defined in (6). Proof.
The key is to show the following estimate, Given a bounded sequence {f n } in E R , by Arzelà-Ascoli Theorem, we get a uniformly convergent subsequence still denoted as {f n }. Then by (15), we see that {Af n } must be a Cauchy sequence in E R . Hence A must be compact in E.
Estimate (15) directly follows from Proposition 1.1 in [27]. For completeness and the convenience of the readers, we prove (15) by a direct computation with the Green's function G R (x, y) in (8). where Second step. Let's show that Af is Hölder continuous in the neighborhood of where Then Hence, Therefore, Af is Hölder continuous in the neighborhood of ∂B R , and Third step. Given any two points x 1 , x 2 ∈ B R , let δ 1 = dist(x 1 , ∂B R ), δ 2 = dist(x 2 , ∂B R ) and δ = min{δ 1 , δ 2 }. There are three cases: For case (i), the estimate (16) holds on Thus, (15) is proved. Corollary 1. Operators A and T = (T 2 , T 1 ) defined in (9)-(10) are compact in E R , which is defined in (7).
Proof. By Theorem 2.4, A is compact in E. So operator (A, A), which is still denoted as A, is compact in E R = E × E.
Due to q > 1, we have g q s,Ω ≤ g q−1 L ∞ (Ω) g s,Ω for g ∈ E. Since T 1 (g) = Ag q , we have and denote Recall λ 1 is the first eigenvalue of (12), so (I − λ 1 A)w = 0 and λ 1 is also called a characteristic value of A. Below we quote the Rabinowitz Theorem (see [1,23,25]), Theorem 2.5. Let A ∈ L(X) be compact and let T ∈ C 1 (X, X) be compact and such that T (0) = 0 and T (0) = 0. Suppose that λ 1 is a characteristic value of A with odd multiplicity. Let C be the connected component of Σ containing (λ 1 , 0). Then either (i) C is unbounded in R × E R , or (ii) there exists another characteristic value of A, λ 2 , such that λ 2 > λ 1 and (λ 2 , 0) ∈ C.
The following lemma states that C only branches out from a bifurcation point to either the positive quadrant, E + R , or the negative quadrant, E − R . So, we only need to consider positive solutions.
Lemma 2.6. Let C be the bifurcation branch of S λ at (λ 1 , 0) defined in Theorem 2.5. Then The proof of the Lemma is almost the same as Lemma 2.6 in [11] and Theorem 3.6 in [12], which needs a maximum principle for the fractional Laplacian (see Theorem 2.8). By this Lemma, hereinafter, we assume u, v > 0, and thus |u| q−1 u = u q and |v| p−1 v = v p . Now, we are going to show that the second case of Theorem 2.5 does not happen for S λ . Indeed, suppose (λ, w) ∈ C∩Σ is a solution to (11), we show that 0 < λ < λ 1 . Due to the definition of λ 1 , λ can not be another characteristic value of A. Thus case two is ruled out.
Proof. Notice that (λ, w) is also a solution to (5), then we can find the associated Pohožaev identity.
Moreover, (u, v) satisfies the condition of Proposition 1.6 in [28]. Now let w 1 = u + v, w 2 = u − v and ν be the unit outward normal vector. By Proposition 1.6 of [28], we have Summing up these two identities, we get Then we use (5) to substitute (−∆) s u, (−∆) s v and get the left side as,
Proof. Again, (λ, w) is also a solution to (5). By Lemma 2.7, (−∆) s u, (−∆) s v > 0 in B R . Meanwhile, u, v ≡ 0 on R n \ B R , and hence by Hopf Lemma for fractional Laplacian (cf. [14]), u δ s , v δ s ≥ c > 0 on ∂B R . Let φ 1 be the first eigenfunction of (−∆) s on B R . It follows that tφ 1 < min{u, v} in B R for small t > 0. Therefore, there exists a t 1 > 0 such that t 1 φ 1 ≤ min{u, v} and Now, if λ ≥ λ 1 , and without loss of the generality, suppose t 1 φ 1 (x 0 ) = u(x 0 ), then we have By the maximum principle (Theorem 2.8), u ≡ t 1 φ 1 on B R , which is impossible since they satisfy different equations.
Thus, we are prepared to prove the first part of Theorem 2.3, Proof of Part I of Theorem 2.3. Due to 0 < λ < λ 1 (by the previous two lemmas) and the simplicity of λ 1 , C must be unbounded in Σ. Moreover, C must be unbounded in R × C(B R ) × C(B R ); otherwise, w ∞ < C, for some C > 0, and by the estimate (15), C must be bounded in Σ, a contradiction. It follows that there exists a solution (λ, w) ∈ C such that w L ∞ (R n ) = u L ∞ (R n ) + v L ∞ (R n ) = 1.

Monotonicity.
For classical elliptic problems, the monotonically decaying property usually is easy to get when the solution is radial. However, this is not the case for problems with fractional Laplacian due to the lack of locality. To show the solution monotonically decays from the origin, we use a direct method of moving plane recently developed by Chen et al. [5], and the key is a maximum principle on a narrow region for some antisymmetric function. Let Theorem 2.10.
Let Ω be a bounded narrow region in Π t , such that it is contained in with small δ. Suppose that w ∈ C 1,1 loc (Ω) and assume it makes sense to take (−∆) s on w and w is lower semi-continuous on Ω. If c(x) is bounded from below, i.e.
then for sufficient small δ = δ(n, s, Λ), we have Furthermore, if w(x) = 0 at x in Ω, then w(x) = 0 almost everywhere in R n .
Remark 2. In [5], δ is not explicitly specified to be independent of t, but indeed it is and δ = δ(n, s, Λ). This can be seen from their proof, and δ is determined by C n,s /δ 2s + Λ > 0. Now we are ready prove the second part of Theorem 2.3.
Proof of Part II of Theorem 2.3. Let . We start to move the plane at t = −R, and since u( We are going to show t 0 = 0 and w t0 We claim that on B R ∩ Π t0−ε , w t0 1 (x), w t0 2 (x) ≥ c for some c > 0. Then by continuous dependence of w t 1 (x) and w t 2 (x) on t, for sufficiently small ε 1 > 0 we have

ZE CHENG, CHANGFENG GUI AND YEYAO HU
Thus, choose ε < min{ε 1 , δ 2 } where δ is from Theorem 2.10, and apply Theorem 2.10 on Ω = (Π t0+ε \ Π t0−ε ) ∩ B R , and then we have w t 1 (x), w t 2 (x) ≥ 0 on Π t , for all t ∈ [t 0 , t 0 + ε]. This is a contradiction to the definition of t 0 . Now we prove the claim. Suppose Using the fact that w t0 1 (x) = −w t0 1 (x t ) and by direct computations we get The last inequality is due to w t0 1 (x) ≥ 0 on Π t0 and w t0 1 (x) = 0 for some x ∈ Π t0 . To see the latter fact, notice that u(x) > 0 in B R and t 0 < t, and let's call the mirror image of B R on the right side of ∂Π t0 , B t0 R . Then in B t0 We get a contradiction, and the claim is proved.
Since t 0 = 0 and w i (x) = w i (|x|), we have obtained the monotonic decaying property of the solution due to the definition of t 0 .
3. Proof of Theorem 1.1. In order to have the desired convergence in Hölder space, we need a local regularity. Proposition 1. Assume that u is a solution to (−∆) s u = f in B R with u ∈ C 0,s (R n ), f ∈ C 0,τ (R n ) and s, τ ∈ (0, 1). Then for any l ∈ (0, R), This estimate is classic and readers may refer to the proof of Proposition 2.8 in [34].
Proof of Theorem 1.1. Let {R j } → ∞ as j → ∞. By Theorem 2.2, there exists a solution to (5) on each B Rj , (λ j , w j ) ∈ (0, λ 1 j ) × E Rj defined in (7), such that Fixing R 0 > 0, then for all j's such that R j > R 0 , we have w j ∈ C 3s (B l ) for any l < R 0 by Proposition 1. A bootstrapping argument allows us to claim that w j ∈ C k,β (B 1 2 R0 ) for any integer k > 0 and β ∈ (0, 1). By Arzelà-Ascoli theorem, we obtain a subsequence of {w j } converging in C k,β (B R0/2 ).
Let R 0 → ∞, and by a diagonal argument we can pick a subsequence of {w j } converging to w = (u, v) ∈ C ∞ (R n ) × C ∞ (R n ) and w L ∞ (R n ) = u(0) + v(0) = 1. The radial symmetry and monotonic decay of w can be easily seen from the radial symmetry and the monotonic decay of u j , v j . Now we need to show w is a solution to (3).
First we show that (−∆) s u j converges to (−∆) s u pointwise. Notice that u j 's are uniformly bounded and indeed |u j (x)| ≤ |u j (0)| ≤ 1. Then for a fixed point x, for any ε > 0 there exists a R > 0 such that Since ε > 0 is arbitrary, by the convergence of {u j } in C k,β (B R (x)) for any k and β ∈ (0, 1), the limit of the last term is zero. Thus, (−∆) s u j → (−∆) s u pointwise. Also, since λ 1 j → 0 as R j → ∞ by a scaling argument, we see λ j u j + v p j → v p . A similar argument follows for the second equation of (3). Therefore, (u, v) is a classic solution to (3).
4. Asymptotic analysis. First, let us show a simple integration by parts lemma. Notice that for a smooth function u with certain decay, e.g., u ∈ H 2s (R n ), the integration by parts holds true due to Fourier transform and Parseval's identity. However, the solution we found may not have such decaying property, and we show the following formula holds.
Proof. First we assume g ∈ C ∞ 0 (R n ) and supp(g) ⊂ B R . Let η ∈ C ∞ be the cutoff function on B 1 , such that η ≡ 1 on B 1 and η ≡ 0 on R n \ B 2 . Let η j (x) = η( x Rj ), and pick a a sequence {R j } that goes to infinity. Let u j = η j u, hence |u j | ≤ |u|.
By Parseval's formula, Now, by the dominated convergence theorem the last term on the right converges to the right of (24). We only need to show the left term above converges to the left term of (24). For sufficiently large j, η j ≡ 1 on B R+1 , then is integrable on B R × R n \ B R+1 , by the dominated convergence theorem the limit of the integral above is 0. Since C ∞ 0 (R n ) is dense in W 2s,1 (R n ), the lemma is proved.
The following energy estimates are similar to estimates for the classical Laplacian for which readers can refer to [9,29] for general 0 < p, q < ∞, and [21,24] for p, q ≥ 1.
Theorem 4.2. For p, q > 1, a solution (u, v) to system (3) must satisfy and where C is a constant depending only on n, s. As a consequence, if (u, v) is a radial and monotonically decaying solution, then u ≤ C|x| −α and v ≤ C|x| −β for some C > 0.
Thus, due to (−∆) s φ 1 < 0 on R n \ B 1 we have, where the last inequality is due to Jensen's inequality. Similarly, Hence, Since pq > 1 and φ 1 ≥ c > 0 on B 1 2 , the above estimate implies (29). Similarly, we get the estimate for v.
As a consequence, a radial and monotonically decaying solution of (3) must decay to zero at infinity and u ≤ C|x| −α and v ≤ C|x| −β for some C > 0 with α, β given by (26).