Qualitative properties of positive solutions for mixed integro-differential equations

This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation \begin{equation}\label{eq 1} \left\{ \arraycolsep=1pt \begin{array}{lll} (-\Delta)_x^{\alpha} u+(-\Delta)_y u+u=f(u)\quad \ \ {\rm in}\ \ \R^N\times\R^M, u>0\ \ {\rm{in}}\ \R^N\times\R^M,\ \ \quad \lim_{|(x,y)|\to+\infty}u(x,y)=0, \end{array} \right. \end{equation} with $N\ge 1$, $M\ge 1$ and $\alpha\in (0,1)$. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.

1. Introduction. The study of qualitative properties of positive solutions to semilinear elliptic equations in R N has been the concern of numerous authors along the last several decades. The asymptotic behavior of the solution at infinity, the actual rate of decay and symmetry properties have been the most studied qualitative properties for these equations. It was the seminal work by Gidas, Ni and Nirenberg [20] that settled these two main qualitative properties for the semi-linear elliptic equation for certain constant c > 0. After this work, many authors extended the results in various directions, generalizing the non-linearity, the elliptic operator or the hypotheses on the solutions. Out of the very many contributions in this direction we mention here only a few: Berestycki and Lions [5], Berestycki and Nirenberg [6], Brock [7], Busca and Felmer [8], Cortázar, Elgueta and Felmer [13], Da Lio and Sirakov [14], Dolbeault and Felmer [16], Gui [21], Kwong [22], Li and Ni [25] and Pacella and Ramaswamy [27].
Recently, much attention has been given to the study of elliptic equations of fractional order. In this direction, Felmer, Quaas and Tan in [18] studied the problem They proved existence and regularity of positive solutions, and also decay and symmetry results. Precisely, it was proved that the solutions u of (1.4) satisfy for some c > 1, when f is superlinear at 0 in the sense that lim s→0 f (s) s = 0.
The radial symmetry of the solutions of (1.4) is derived by using the moving planes method in integral form developed in [11,26], assuming further that f ∈ C 1 (R), it is increasing and there exists τ > 0 such that lim s→0 f (s) s τ = 0. (1.6) This symmetry result was generalized by the authors in [19], using an appropriate truncation argument together with the moving planes method with ideas developed in [24]. We refer to some other papers with more discussions on qualitative properties of solutions to fractional elliptic problems as Cabré and Sire [9], Caffarelli and Silvestre [10], Chen, Li and Ou [11], Barles, Chasseigne, Ciomaga and Imbert [2], Dipierro, Palatucci, Valdinoci [15], Li [26], Quaas and Xia [29], Ros-Oton and Serra [30], Sire and Valdinoci [33]. Both operators, the laplacian and the fractional laplacian, are particular cases of a general class of elliptic operators connected to backward stochastic differential equations associated to Brownian and Levy-Itô processes, see for example Barles, Buckdahn and Pardoux [1], Benth, Karlsen and Reikvam [4] and Pham [28]. Recently, Barles, Chasseigne, Ciomaga and Imbert in [2,3] and Ciomaga in [12], Esfahani and Esfahani in [17] considered the existence and regularity of solutions for equations involving mixed integro-differential operators belonging to the general class of backward stochastic differential equations mentioned above. A particular case of elliptic integro-differential operator of mixed type is the one considering the laplacian in some of the variables and the fractional laplacian in the others, modeling diffusion sensible to the direction. In view of (1.1) and (1.4) we may write similarly (−∆) α x u + (−∆) y u + u = f (u), (x, y) ∈ R N × R M , u > 0 in R N × R M , lim |(x,y)|→+∞ u(x, y) = 0, (1.7) where N ≥ 1, M ≥ 1. The operator (−∆) y denotes the usual laplacian with respect to y, while (−∆) α x denotes the fractional laplacian of exponent α ∈ (0, 1) with respect to x, i.e.
for all (x, y) ∈ R N × R M . Here the integral is understood in the principal value sense.
In view of the known results on decay and symmetry for solutions of equations (1.1) and (1.4) just described above, it is interesting to ask if these results still hold for solutions of the equation of mixed type (1.7), where the elliptic operator represents diffusion depending on the direction in space. Regarding the asymptotic decay of solution at infinity, the question is interesting since a proper mix of the two variables should be obtained for the decay estimates. The natural way to estimate the decay is through the construction of super and sub solutions involving the fundamental solution of the elliptic operator, which in this case is singular in R N × {0}. Moreover, the solution of (1.7) cannot be radially symmetric, so this property cannot be used to estimate the decay. On the other hand, regarding radial symmetry, we may still have symmetry in x and y, but the moving planes method would require an adequate version of the Hopf's Lemma, that we prove here.
Our first theorem concerns the decay of solutions for (1.7) with general nonlinearity and it states as follows.
Theorem 1.1. Let α ∈ (0, 1), N, M ∈ N, N ≥ 1 and M ≥ 1 and let us assume that the function f : (0, +∞) → R is continuous and it satisfies Let u be a positive classical solution of problem (1.7), then for any > 0 small, there exists C > 1 such that for any (x, y) ∈ R N × R M , where When we compare estimate (1.10) with (1.3) for N = 0, we first observe that in ours an exponential decay is obtained, but with a constant C depending on , which is a parameter controlling the rate of exponential decay. This is more clear when A = B = 0. On the other hand we are making much more general assumptions on f and, in particular, we are not making any assumption on the radial symmetry of the solution, which is crucial in proving (1.3). We do not know of a decay estimate better than Assume that α ∈ (0, 1), N ≥ 1, M ≥ 5 and the non-linearity f : (0, +∞) → R is non-negative and it satisfies (1.2). Let u be a positive classical solution of (1.7), then there exists a constant c > 1 such that for all (x, y) ∈ where the function ρ is defined as (1.14) We notice that this theorem gives the expected exponential decay for positive solutions, as suggested by (1.3), assuming the dimension of the space satisfies M ≥ 5. Moreover, it gives the expected polynomial correction for the lower bound with a gap in the power for the upper bound. This theorem is proved under the assumption (1.2) on the non-linearity, constructing super and sub solutions devised upon the fundamental solution of (−∆) α x + (−∆) y + id. In our argument, a crucial role is played by the estimate already obtained in Theorem 1.1. Since the fundamental solution of (−∆) α x + (−∆) y + id has R N × {0} as singular set, we cannot use the method in [20] in order to derive our estimate. Moreover, some other arguments in [20] cannot be used either because the solutions of (1.7) are not radial, since the differential operator is not radially invariant and there are no solutions depending only on one of the x or y variables, as can be seen from (1.13), Even though solutions of (1.7) are not radially symmetric, we can prove partial symmetry in each of the variables x and y and this is the content of our third theorem. Theorem 1.3. Assume that α ∈ (0, 1), N ≥ 1, M ≥ 1 and the function f : (0, +∞) → R is locally Lipschitz and it satisfies (1.9). Moreover, we assume that f also satisfies When N = 0, we see that assumption (F ) implies γ > 0 and (1.15) coincides with the assumption considered in [24]. When M = 0, assumption (F ) implies that γ > 2α N +2α and it coincides with the assumption considered in [19], when the solutions is assumed to decay as a power N + 2α at infinity. We remark that the operator (−∆) α x +(−∆) y is a combination of two operators with different differential orders in x−variable and y−variable, and this produced a combined polynomial-exponential decay and does not allow for radial symmetry, but only partial symmetry as stated in Theorem 1.3.

QUALITATIVE PROPERTIES OF POSITIVE SOLUTIONS 373
The proof of Theorem 1.3 is based on the moving planes method as developed in [19,24]. In these arguments, the strong maximum principle plays a crucial role and it is available for the laplacian and for the fractional laplacian. However, in the case of our mixed integro-differential operator some difficulties arise and we overcome them with a version of the Hopf's Lemma.
The rest of the paper is organized as follows. In Section §2, we introduce a version of the Hopf's Lemma and a strong maximum principle. In Section §3, we prove the decay of solutions as in Theorem 1.1 and Theorem 1.2 by constructing suitable super and sub solutions. Section §4 is devoted to prove symmetry results presented in Theorem 1.3.

2.
Preliminaries. This section is devoted to study the Strong Maximum Principle for mixed integro-differential operators as in equation (1.7). To this end, we prove first a suitable form of the Hopf's Lemma.
However, before to go to this, we recall some basic properties of the Sobolev embeddings. If we denote the Sobolev spaces respectively, then we have following inclusions. Proposition 2.1. For α ∈ (0, 1), we have that where the first inclusion is continuous and the second inclusion is continuous if Proof. Since |ξ 1 | 2α + |ξ 2 | 2 + 1 ≥ |ξ| 2α +1 2 , we have that the inclusion: H(R N +M ) ⊂ H α (R N +M ). From Lemma 2.1 in [18], it shows that the inclusion: N +M −2α and the inclusion H(R N +M ) ⊂ L p loc (R N +M ) is compact if 1 ≤ p < 2(N +M ) N +M −2α . We devote the rest of this section to prove the Strong Maximum Principle in our context and to this end, we start with versions of the Maximum Principle and the Hopf's Lemma. In what follows, given Ω an open subset in R N × R M , we define its closed cylindrical extension in the direction x as Given a function h defined in an appropriate domain, we consider the mixed integrodifferential operator Proof. If not, we may assume that there exists some (x 0 , y 0 ) ∈ Ω such that and (−∆) y w(x 0 , y 0 ) ≤ 0 and then, since h is non-negative we have Lw(x 0 , y 0 ) < 0, which contradicts (2.1), completing the proof.
It what follows we prove a version of the Hopf's Lemma and for this purpose we need to give some conditions to the boundary of the domain where the function is defined. We say that the domain Ω ⊂ R N × R M satisfies interior cylinder condition : |y −ỹ| < r} and, obviously |ỹ − y 0 | = r. We define also ∈ Ω. Further assume that for r > 0 is given in (2.3) and for any (x, y) ∈ D, we have that where n is the unit exterior normal vector of Ω at the point (x 0 , y 0 ).
Proof. Let us define where β > 0 will be chosen later. By direct computation, we have that where ϕ N is the first eigenfunction of Dirichlet problem where ϕ N is positive and bounded in B N r/2 (x 0 ) and the first eigenvalue λ 1 , is positive, see Propositions 9 and 4 in [31] and [32], respectively.
For (x, y) ∈ D, by (2.8) and (2.9), we obtain that where the last inequality holds by the fact that 0 ≤ ϕ M (y) < e −β|y−ỹ| 2 and |y −ỹ| > r/2 in D. Let us choose β > 0 big enough such that On the other hand, since ϕ N (x) = 0 for |x−x 0 | ≥ r/2 and ϕ M (y) = 0 for |y −ỹ| = r, it is obvious that We also observe that v is a bounded function inÕ r . Next we prove (2.5) assuming h ≥ 0. Defining and using (2.4), we have that for any (x, y) ∈ D, Combining with (2.10), we have that, for every > 0 (2.12) Since v is bounded inÕ r , the set A 3 is a compact subset of O r and w > 0 in O r , then there exists > 0 small such that In view of the definition of W , since D ⊂Ō r , we find that w − v ≥ 0 in D and noticing that w(x 0 , y 0 ) = v(x 0 , y 0 ) = 0 we obtain that for all s ∈ (0, r/2). Thus, we have completing the proof of (2.5).
The case for general h can be done simply by replacing h by h + . In fact, since w > 0 in Ω, we have ∈ Ω and similarly we obtain that ∈ D, so we may proceed as before to get (2.5) and the proof is complete.
In order to state the Strong Maximum Principle to be used in our moving planes procedure, it is convenient to consider property (P ): (P ) We say that a function w :Ω → R satisfies property (P ) if whenever (x 0 , y 0 ) ∈ Ω such that The following lemma is in preparation of the strong maximum principle.
Then w must be 0 inΩ.
3.1. Proof of Theorem 1.1. In this subsection, we prove Theorem 1.1 on decay estimates for positive classical solutions of equation (1.7). The main work is to construct appropriate super and sub solutions and then the decay estimate is derived by Lemma 2.1. Before proving Theorem 1.1, we introduce some computations gathered in the next proposition. For α ∈ (0, 1) and µ > 0, we define the function ψ µ : R N → R as follows Proof. We consider along the proof that µ > 0 and x ∈ R N satisfies |x| > 3µ. We define and we observe that Now we compute the integral above by decomposing the domain in various pieces. First we consider the integral over B |x| 4) where e x = x |x| and c 1 , c 2 > 0 are independent of µ. Next we consider the integral over B |x| where the first inequality holds since |z For the inequality on the other side, we obtain where the second inequality holds by |z| ≤ 4 3 for z ∈ B 1 3 (e x ). Consequently, 5) where the constants c 4 , c 7 , c 8 > 0 are independent of µ. The estimate for the integral over B |x| Next we consider the integral over B µ (x). We observe that, for z ∈ B µ (x) we have since |x + z| > µ > |x − z| and |z| ≥ |x| − µ ≥ 2|x| 3 , thus and, for the other inequality where c 9 , c 10 , c 11 and c 12 are positive constant independent of µ. Therefore, The integral over B µ (−x) is exactly the same. Finally, we consider the complemen- (−x)). For |x| > 3µ and z ∈ D(x), we have that |x ± z| ≥ |x| 3 , thus where c 13 > 0 and c 14 > 0 are independent of µ. Therefore, by (3.4)-(3.7), there exist c 15 , c 16 > 1 independent of µ such that In what follows we provide a proof of our first theorem on the decay of the positive solutions of our equation.

PATRICIO FELMER AND YING WANG
Combining (3.10) with (3.24), we obtain that (0)). By Lemma 2.1, we have then Step 6. There exist C 1 ( ) > 0 such that, for R as in Step 4, To prove this we letṼ (x, y) = ψ µ (x)φ θ2 (y), for (x, y) ∈ R N × R M with µ as defined above. Using (3.2) and (3.11), for ( Since ψ µ is bounded in B N R (0), using (3.21), there existsr 2 > 0 such that u −r 2 V ≥ u −r 2 c 1 e −θ2|y| ≥ 0 in B N R (0) × R M , and by (3.22), there existsr 3 > 0 such that . Takingr = min{r 1 ,r 2 ,r 3 } and combining (3.10) with (3.26), we obtain that Our proof is based on the fundamental solution of the mixed integro-differential operator. We first study the fundamental solution K for In fact, for φ ∈ S, we have that Next we want to find some properties of H. To this end, we consider It is well known that the function H α has the following properties: where C > 0, which imply that there exists c 1 > 0 and c 2 > such that see [23,18]. By the definition of H, we have that Since we have see [23], together with (3.27)-(3.30), for |y| > 2, for some c 3 > 0. On the other hand, since for n ≥ 3 we have with c 4 > 0 (see [23]), for M ≥ 5 we have that where ρ(x, y) is defined in (1.14). In what follows, we construct super and subsolutions to obtain the decay estimate given in Theorem 1.2.
Proof of Theorem 1.2. By the estimate in Theorem 1.1, we observe that, for constants c 10 > c 9 > 0 such that ) and, by (3.32) and Theorem 1.1, there exists c 12 > 0 such that u ≥ c 11ũ in R N ×{y ∈ R M : |y| = 2}. Since f is nonnegative, we use the Comparison Principle to obtain that, for any ( Step 2. Upper bound. For y ∈ R M with |y| ≥ 2, there exists 1 ≤ i ≤ M such that |y i | > 1, we may assume that y 1 > 1. Letū(x, y) = K(x, y)(1 − |y 1 | −1 ), then by direct computation , where the last inequality holds since y 1 > 0 and ∂ y1 K < 0. Therefore, by (3.32), we have that for (x, y) Since f (u) = O(u p ) near u = 0 for some p > 1, by Theorem 1.1 with = p−1 4p , we have that where c 13 > 0. We notice that 3p+1 4. Symmetry results. In this section, we prove Theorem 1.3 by moving planes method. Let u be a classical positive solution of (1.7) and consider first the ydirection. Let We introduce a preliminary inequality which plays a crucial role in the procedure of moving planes.
Proof. First we show that the integrals are finite. We observe that u λ satisfies the same equation (1.7) as u in Σ y1 λ . Taking (u λ − u) + as test function in the equations for u and u λ , subtracting and integrating in Σ y1 λ , we find Now we only need to prove that In fact, for any given λ ∈ R, using (1.10), we choose R > 1 such that 0 < u λ (x, y) ≤ C (1 + |x|) −N −2α e −θ1|y λ | < s 0 , ∀(x, y) ∈ B c R , where y λ = (2λ − y 1 , y ) for y = (y 1 , y ) ∈ R M and s 0 is from (F ).
If u λ (x, y) > u(x, y) for some (x, y) ∈ Σ y1 λ ∩ B c R , we have 0 < u(x, y) < u λ (x, y) < s 0 . Using (1.15) with v = u λ (x, y), then The inequality above is obvious if u λ (x, y) ≤ u(x, y) for some (x, y) ∈ Σ y1 λ ∩ B c R . Then where the last inequality holds by γ > 2αN (N +M )(N +2α) . Since u and u λ are bounded and f is locally Lipschitz, we have Therefore, (4.2) holds. Together with (4.1), we have the second inequality in the result.
Proof. We divide the proof into three steps.
Step 1. λ 0 := sup{λ | u λ ≤ u in Σ y1 λ } is finite. Since u decays at infinity, we observe that the set {λ | u λ ≤ u in Σ y1 λ } is nonempty. Using (u λ − u) + as a test function in the equation for u and u λ , by (1.15) and Hölder inequality, for λ big (negative), we find that . Since γ > 2αN (N +2α)(N +M ) , we have that a > N . Then we can choose R > 0 such that for all λ < −R, By Lemma 4.1, we obtain that Thus u λ ≤ u in Σ y1 λ for all λ < −R and then conclude that λ 0 ≥ −R. On the other hand, since u decays at infinity, then there exist λ 1 ∈ R and (x, y) ∈ Σ y1 λ such that u(x, y) < u λ1 (x, y). Hence λ 0 is finite.
Step 2. u ≡ u λ0 in Σ y1 λ0 . Assuming the contrary, we have that u ≡ u λ0 and u ≥ u λ0 in Σ y1 λ0 , in this case the following claim holds. Claim 1. If u ≡ u λ0 and u ≥ u λ0 in Σ y1 λ0 , then u > u λ0 in Σ y1 λ0 .
Step 3. By translation, we may say that λ 0 = 0. Repeating the argument from the other side, we find that u is symmetric about y 1 -axis. Using the same argument in any y-direction, we conclude that u(x, y) = u(x, |y|), (x, y) ∈ R N × R M .
Using the result of u(x, y) = u(x, |y|) for all (x, y) ∈ R N × R M and | y 1 | < |y 1 |, we conclude monotonicity of u respect to y. This completes the proof.
Next we study the symmetry result in x-direction. and u is strictly decreasing in x-direction.