Monotonicity formulas for coupled elliptic gradient systems with applications

Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-\Delta)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas and Spruck in \cite{cgs} provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is when $m\ge1$ and $s=1$ a similar classification result is given by Druet, Hebey and V\'{e}tois in \cite{dhv} and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, when $s=1,2$, and nonlocal, when $0<s<1$ and $1<s<2$, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman \cite{acf} seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.


Introduction . Background
Our objective in this paper is to establish monotonicity formulae for solutions of the following coupled elliptic system of equations (−∆) s u i = |u| p− u i in R n (1.1) and classify nite Morse index solutions when < s ≤ , p > , m ≥ , u = (u i ) m i= and u i : R n → R. Note that when m = the above is known as the Lane-Emden equation. An important feature of system (1.1) is the variational structure of the coupled equations of the form −∆u = ∇H(u) in R n . (1.2) Note that (1.1) is a particular case of (4.33) for H(u) = p+ |u| p+ . Due to this variational structure, qualitative behaviour of solutions of system (4.33) has been studied extensively in the context of elliptic partial di eren-tial equations from both pure and applied mathematics perspectives. Let us mention that Andersson et al. in [4] considered system (1.1), up to a negative sign, when p = and s = which reads This is closely related to minimizers of the energy m i= |∇u i | + |u| dx, (1.4) and they studied the regularity of free boundaries. To prove regularity results in [4], they established a monotonicity formula that is inspired by the one given by Weiss in [44,45]. We also refer interested readers to [3,7] for regularity results on cooperative systems and to the book of Petrosyan, Shahgholian and Uraltseva in [36] for more information. It should be remarked that as a particular case, one can consider the case of two components, that is m = , and p = namely (1.5) Note that the above system is a special case of the nonlinear Schrödinger system that is where µ , µ , β are constants. We would like to mention that most of the results in the paper are valid for (1.6) as well. We refer interested readers to [34,41,42] and references therein for more information regarding classi cation of solutions of system (1.6) for various parameters µ , µ and β.

. Tools and methodology (Historical development)
Semilinear elliptic equations (with almost four decades of history) are probably the most widely and intensely studied equations in PDE. The simplest equation in this class is expressed as ∆u = f (u), (1.7) and exhibits as many possible features, as the right hand side nonlinearity f (u) may do. This equation has also been studied from so many di erent perspectives, that any attempt to try to list them here would probably fail. There are however two di erent types of questions, of interest to the current authors, that seem to be interconnected and developed in parallel, but almost separately: i) Analysis of the singular set: {x : ∇u = }. ii) Stability and Liouville's type questions.
One can actually add a third less studied, and probably more complicated problem to the above, which is: iii) Structure of the blow-up set: {x : |u(x)| = ∞}.
In studying the above problems, experts have (in many cases) developed parallel tools and ideas to handle technical passages of the analysis of questions in each cases. Two of these tools, also utilized in this paper, are Monotonicity functional, and Blow-up/down Analysis.
The particular monotonicity functional used here combines the energy functional along with a balanced term (see below for explicit form). The use of monotonicity functional (in the way we present here) can be traced back¹ to the work of Fleming in [25] for area minimizing currents which has subsequently been proved by Allard in [2] for stationary recti able n−varifolds, that was later developed further by others. The monotonicity functional that we shall use in this paper originates in the study of harmonic maps by Price in [37] and Schoen and Uhlenbeck in [40], see also [39]. And in connection to free boundary problems, it was rst used by Ou [5], and developed later by Weiss in [45] and others in various forms. In the context of semilinear heat equations, a similar monotonicity formula is given by Giga and Kohn in [28] and for the corresponding elliptic equation by Pacard [35].
The second main tool, blow-up/down analysis, has its origin in local regularity theory and the so-called linearization technique (also called harmonic blow-up). To study local structure of level surfaces of a solution to the above semilinear problem, one considers scaled version of the problem and classi es the limit manifold of such scalings. The limit manifold naturally carries information of the local behavior of the solutions and hence one can then with some devices link back this behavior to the local problem and deduce the expected properties. It is now that the role of monotonicity functional become crucial and indispensable in that one can use the fact that when blowing up a solution, the monotonicity functional, call it E(r, u), being monotone will have a limit. At the same time this functional has a nice scaling properties E(rs, u) = E(s, ur) for r, s > and ur a correctly scaled version of u. Hence one obtains so that E(s, u ) is constant. Now a strong version of the monotonicity functional asserts that the only time E is constant is when the function u is homogeneous, where the order of homogeneity is dictated by E. Since the same can be done by blowing-down, we will have that u , u∞ are homogeneous of same order. Next if one can prove that the homogenous solutions are unique we must then have u = u∞ and hence E( + , u) = E(∞, u), and we arrive at E(r, u) = constant. Therefore u is itself homogeneous, and already classi ed.
In the regularity theory of level surfaces the classi cation of homogeneous global solutions is one of the key elements of the theory, and many time a very hard nut to crack, in higher dimensions; in two dimensions homogenous solutions can be computed directly. In the stability theory and Liouville type problems this part is based on computations of eigenvalues of the corresponding Laplace-Beltrami on the sphere, as also done in this paper. As shown here below (and in many preceding papers by experts in this area) the only homogenous solutions in the appropriate space are the trivial solutions. In particular, the energy functional becomes zero and hence E(r, u) = . From here one deduces that u ≡ is the only solution satisfying the given condition.
It is worthwhile remarking the fundamental feature of these problems, that in some cases depend on the dimension and order of homogeneity are directly a consequence of the eigenvalues of Laplace-Beltrami, where in the case of index-theory results in the computation of eigenvalues for nonlinear Laplace-Beltrami, that in turn implies that non-trivial solutions do not exists in certain range of values. These ideas have their origin in the regularity theory of minimal surfaces, that boils down to proving minimal cones do not exists in dimensions less than eight.

. Problem setting
Our main technique is to derive monotonicity formulae for solutions of (1.1) for various values of parameter < s ≤ . To provide such monotonicity formulae we consider various cases. Case s = . Consider the following functional for every λ > and x ∈ R n |u| , (1.9) then the following monotonicity formula holds for classical solutions of (1.1).
that for a jordan curve γ, the product of the harmonic measures for both sides of the curve can be controlled as follows: ω (Br(z) ∩ γ) · ω (Br(z) ∩ γ) ≤ Ar , where z ∈ γ, and ω , ω denote the harmonic measures on each side of the curve. where ∂ ∂r is polar derivative.
Note that for the case of single equations, that is when m = , similar monotonicity formulae are given by Pacard in [35] and Weiss in [44,45]. For the case of systems that is when m ≥ , very recently, Andersson, Shahgholian, Uraltseva and Weiss in [4] provided a monotonicity formula for solutions of system (1.1) when p = and applied it to study free boundary problems. Let us mention that very similar monotonicity formulae appear in the eld of harmonic maps that is −∆u = |∇u| u, (1.11) where u : R n → S m− , see Evans in [18,19] and reference therein.
Then the following monotonicity formula holds for the fourth order Lane-Emden system.

Theorem 1.2. Suppose that n ≥ , p > n+ n− and u is a solution of (1.1) when s = . For any λ > and x
where E is de ned by (1.12) and C is independent from λ.
Note also that the above monotonicity formula is provided by Davila, Dupaigne, Wang and Wei in [15] for the case of single equations that is when m = . For the case of fractional Laplacian, we provide monotonicity formulae for the extension problems. Assume that u i ∈ C σ (R n ), σ > s > and for each ≤ i ≤ m. The fractional Laplacian of u i when < s < denoted by is well-de ned for every x ∈ R n . Here p.v. stands for the principle value. It is by now standard that the fractional Laplacian can be seen as a Dirichlet-to-Neumann operator for a degenerate but local di usion operator in the higher-dimensional half-space R n+ , see Ca arelli and Silvestre in [8]. In other words, for for the following constant κs, For the case of < s < , there are various de nitions for the fractional operator (−∆) s , see [9,12,24,47]. From the Fourier transform one can de ne the fractional Laplacian by . Note that Yang in [47] gave a counterpart of the extension problem by Ca arelli and Silvestre in [8] for the fractional Laplacian (−∆) s , where s is any positive, noninteger number. In other words, he showed that the higher order fractional Laplacian operator can also be regarded as the Dirichlet-to-Neumann map for an extension function satisfying a higher order elliptic equation in the upper half space with one extra spatial dimension. More precisely, there exists an extension function v i ∈ W , (R n+ where b := − s and ∆ b v i := y −b div(y b ∇v i ). We refer interested readers to Case and Chang [9] and Chang and Gonzales [12] as well. We are now ready to provide monotonicity formulae for various parameters s > .
i= be a solution of the extension problem (1.16). Now de ne the energy functional for any λ > and x ∈ ∂R n+ We are now ready to provide a monotonicity formula for solutions of (1.16) when < s < .
Then, Es is a nondecreasing function of λ. Furthermore,
i= is a solution of the extension problem (1.19). Similarly, now de ne the energy functional for any λ > and x ∈ ∂R n+ Considering the above energy functional, we now provide a monotonicity formula for solutions of (1.19) when < s < .
where Es is given by (1.22) and C(n, s, p) is independent from λ.
We also refer interested readers to Davila, Dupaigne and Wei in [14] and to Wei and the rst author in [24] for a similar monotonicity formula for the case of scalar equations and < s < and < s < , respectively. Before we state our main results let us present some backgrounds regarding classi cation of solutions of (1.1) in the absence of stability. We provide such classi cations for scalar equations and systems separately.  see also [33]. Note that for the fractional Laplacian operator < s < , such a classi cation result is given by Li [32] and Chen-Li-Ou [11] where the critical exponent is Then they proved that there exist (1.28) We also refer interested readers to [16,29,30] where the authors studied various counterparts of system (1.1). Note that the following singular function Here is our main result. Theorem 1.5. Suppose < s ≤ and n > s. Let u be a solution of (1.1) that is stable outside a compact set. Then either for < p < p S (n, s) or for p > p S (n, s) and

each component u i must be identically zero. For the case of Sobolev critical exponent, that is when p = p S (n, s), a solution u has nite energy that is
(1.32) In this case, if in addition u is stable, then each component u i must be identically zero.
As a direct consequence, the above theorem implies that the only nonnegative solution for system (1.5) when s = is the trivial solution for dimensions n < and n ≠ . Here, is how this article is structured. For the rest of this article, we provide a proof for Theorem 1.5 considering various cases for parameter s > . In Section 2, we consider the case s = that is when the operator in (1.1) is the local Laplacian operator. In Section 3, we let s = that refers to the bi-Laplacian operator. Lastly, in Section 4, we consider nonlocal cases < s < and < s < . For these non-integer parameters, the operator in (1.1) is a fractional Laplacian operator. For all cases < s ≤ , we apply blow-down analysis arguments and monotonicity formulae.

Local Case: Laplacian operator
In this section we assume that s = . Therefore system (1.1) turns into the following form Note that this is a particular case of system (4.33). For a general nonlinearity H : R m → R such that ∇H ≥ , it is proved in [22] that bounded stable solutions of (4.33) are constant when n ≤ . For radial solutions, this is known to hold in more dimensions that is when n ≤ without any sign assumptions on the nonlinearity, see [21]. In addition it is known that at least for the case of radial solutions the dimension n = is the optimal dimension. For the rest of this section we prove the following classi cation of nite Morse index solutions of (2.1).
Then each u i must be identically zero. For the Sobolev critical exponent p = n+ n− , a solution u has nite energy that is

In this case, if in addition u is stable, then each component u i must be identically zero.
Note that for the case of scalar equations, that is when m = , the above theorem is given by Farina in [20]. He used a Moser iteration type argument for the proof. We refer interested reader to Crandall and Rabinowitz in [13] for a similar approach. To prove the above theorem, we apply a blow-down analysis argument as well as the monotonicity formula given as Theorem 1.1. We now derive a few elliptic estimates. where C is a positive constant that is independent from R.
Proof. Test the stability inequality on . Now multiply both sides of (1.1) with u i ζ R and integrate by parts. Equating the inequalities that one gets from this and from stability nishes the proof.
Applying the Hölder's inequality, we get the following L estimate.

Corollary 2.1. Suppose that u is a stable solution of (1.1). Then, for any R
where C is a positive constant that is independent from R.
In this part, we present a classi cation of stable homogeneous solutions. This is a key point in our proof of Theorem 2.1.

Theorem 2.2. Suppose that u = (u i ) m i= for u i = r − p− ψ i (θ) is a stable solution of (1.1). Then, each ψ i is identically zero provided n+ n− < p < pc(n) where pc(n) is given by (2.2).
Proof. We omit the proof here, since a similar argument will be provided in the proof of Theorem 3.2 for the fourth order case.
Proof of Theorem 2.1. The proof is based on a blow-down analysis and it relies on the monotonicity formula provided as Theorem 1.1. We omit the details, since similar arguments will be provided for the poof of Theorem 3.1. For the case of Sobolev critical exponent, one can conclude the result via applying the Pohozaev indentity.

Local Case: Bi-Laplacian Operator
In this section, we consider the following fourth order system This section is devoted to the proof of the following theorem.
Then, each u i must be identically zero. For the Sobolev critical exponent p = n+ n− , a solution u has nite energy that is

In this case, if in addition u is stable, then each component u i must be identically zero.
In order to prove the above theorem, we are required to establish some a priori estimates on solutions.
Lemma 3.1. Suppose that u = (u i ) m i= be a smooth stable solution of (3.1) and set w = (w i ) m i= where w i = ∆u i . Then the following estimate holds, for a test function ζ : Proof. Test the stability inequality on u i ζ where ζ is a test function then multiply both sides of the i th equation of (1.1) with u i ζ . Equating these completes the proof.
Applying an appropriate test function yields the following estimate.

5)
and therefore Proof. Set the test function ζ R ∈ C c (R n ) in (3.1) where ≤ ζ R ≤ being the following test function We now provide classi cation of stable homogeneous solutions. Note rst that the following Hardy-Rellich inequality with the best constant holds. Suppose that h : This inequality implies that the singular solution given by (1.29) is stable if and only if where α := p− and A given in (1.29).
Proof. It is straightforward to see that ψ = (ψ i ) m i= satis es for α := (q + )(n − − q) + q(n − − q) and β := q(q + )(n − − q)(n − − q), (3.10) where q := p− . Multiplying both sides of (3.9) with ψ i and integrating over S n− , we conclude We now test the stability inequality (1.30) on ϕ = (ϕ i ) m i= for ϕ i := r − n− ψ i (θ)ηϵ(r). Here, ηϵ is a standard cut-o function ηϵ ∈ C c (R+) at the origin and at in nity that is ηϵ = for ϵ < r < ϵ − and ηϵ = for either r < ϵ/ or r > /ϵ. Applying similar ideas provided in [24], we get Combining (3.11) and (3.12) we get Note the coe cients p − , pα − n(n− ) and pβ − n (n− ) are positive when n+ n− < p < pc(n) where pc(n) is given by (3.2). Proof of Theorem 3.1. The proof of the Sobolev critical case relies on applying the Pohozaev identity and we omit it here. We now provide a sketch of the proof when p > n+ n− in a few steps. Step 1. limr→∞ E(u, r, ) < ∞. Note that the energy E(u, r, ) is nondecreasing in r, as given in Theorem 4.14. Therefore, (3.14) Applying estimates given in Corollary 3.1 imply that the right-hand side of (3.14) is bounded. Step From elliptic estimates, up to a subsequence, . From compactness embeddings and applying interpolation we arrive at u λ i → u ∞ i in L q loc (R n ) for any q ∈ [ , p + ). Note also that u ∞ is a stable solution, since u λ is a stable solution and we can send λ to in nity.
Step 3. u ∞ is a homogeneous solution. This is a direct consequence of the monotonicity formula and the following fact lim The left-hand side of (3.16) is bounded from below by (3.18) This implies that for each ≤ i ≤ m, we have This completes the proof of this step.
Step 4. limr→∞ E(u, r, ) = . Since each u ∞ i is a homogenous function, Theorem 3.2 implies that u ∞ = . Therefore, lim λ→∞ u λ i = strongly in L (B ) for each i = , · · · , m that is The fact that E is nondecreasing, that is given as a monotonicity formula in Theorem 1.2, completes the proof.

Nonlocal Case: Lower and higher order fractional Laplacian
In this section, we consider system (1.1) with the fractional Laplacian operator (−∆) s where s ∈ ( , ) for s ≠ and we establish Theorem 1.5. We rst note that the following Hardy inequality holds for n > s for any h ∈ C ∞ c (R n ) where the optimal constant is given by Note that Λ n, = n (n− ) meaning (4.1) recovers (3.7) for s = . For more information interested readers are encouraged to see [29] by Herbst (and also [46]). We now provide a classi cation result for homogeneous solutions. Note that for the case of scalar equations, that is when m = , this classi cation was given in [24]. Note also that the proof is valid regardless of magnitude of the parameter s.
is a stable solution of (1.1) for s ∈ ( , ) and s ≠ . Then, each ψ i vanishes identically, provided p > n+ s n− s and Proof. Since u = (u i ) m i= is a solution of (1.1), each ψ i satis es where we have used the change of variable |y| = t|x|. Simplifying the above, for each i, we obtain where and Multiplying (4.5) with ψ i and integrating we get We now test the stability inequality (1.30) for ϕ i (x) = r − n− s ψ i (θ)ηϵ(r) and u i = r − s p− ψ i (θ) with the same ηϵ(r) as the one given in the proof of Theorem 3.2. Applying similar ideas provided in [24], we conclude Λn,s The fact that Kα is decreasing in α implies K n− s < K s p− for p > n+ s n− s . Therefore, K n− s − pK s p− < . On the other hand the assumption of the theorem implies that Λn,s − pAn,s < . Therefore, each ψ i vanishes identically. This completes the proof.

. Lower Order Fractional Laplacian Operator
In this part, we show that Theorem 1.5 holds when < s < . To do so we provide the following estimate rst. Proof of Theorem 1.5 when < s < . We omit the proof here since arguments are very similar to the ones which will be provided for the case of < s < .

. Higher Order Fractional Laplacian Operator
As the last past of this section, we shall restrict ourselves to the case < s < . Let us start with the following integral estimate on stable solutions.  Note also that for each i these identities hold for a test function η (4.18) Applying these identi es together with (4.16) one can see that Cn,s ∂R n+ Testing the stability inequality (1.30) on ϕ i = u i η and applying (4.19) one can nish the proof.
One can set η to be the standard test function to get the following estimate.
Here we provide more decay estimates of solutions. These lemmata are main tools in our proof of Theorem 1.5. We omit the proofs here and we refer interested readers to see the proof of Lemma 4.5-4.6 in [24] where similar arguments are applied. Proof of Theorem 1.5 when < s < . The proof of the case p = p S (n, s) is based on the Pohozaev identity provided in [24,38], and we omit it here. The proof is based on the monotonicity formula that is Theorem 1.4 and a blow-down analysis.
Step 1. The energy is bounded, that is lim λ→∞ E(v, , λ) < ∞. This is a direct consequence of the monotonicity formula and