GLOBAL BIFURCATIONS AND A PRIORI BOUNDS OF POSITIVE SOLUTIONS FOR COUPLED NONLINEAR

. In this paper, we consider the following coupled elliptic system ∞ . Under symmetric assumptions λ 1 = λ 2 ,µ 1 = µ 2 , we determine the number of γ -bifurcations for each β ∈ ( − 1 , + ∞ ), and study the behavior of global γ -bifurcation branches in [ − 1 , 0] × H 1 r (cid:0) R N (cid:1) × H 1 r (cid:0) R N (cid:1) . Moreover, several results for γ = 0, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6]

1. Introduction. Consider the following doubly coupled nonlinear Schrödinger system where N = 2, 3, µ 1 and µ 2 are positive constants, γ and β are linear and nonlinear coupling constants, respectively. The problem (1) has many applications in physics, especially in nonlinear optics, see [1,3,8,15,24,25,26,31] and references therein. The solutions Φ and Ψ denote the first and the second components of the beam in Kerr-like photorefractive media [1]. The positive constant µ j is for self-focusing in the j-th component of the beam, j = 1, 2. The nonlinear coupling constant β is the interaction between the two components of the beam. The interaction is attractive if β > 0, and repulsive if β < 0. The linear coupling is generated either by a twist applied to the fiber in the case of two linear polarization, or by an elliptic deformation of the fibers core in the case of circular polarizations. Problem (1) also arises in the dynamics of the Bose-Einstein condensates with two degrees of freedom (see [19,27,24,34,30] and references therein for more details). Physically, Φ and Ψ are the corresponding condensate amplitudes, µ j and β are the intraspecies and interspecies scattering lengths. When β < 0, the interactions are repulsive [34]; in contrast, when β > 0, they are attractive. The linear coupling constant γ denotes the strength of the radio-frequency or electric coupling (see [17]). Consider solitary wave solutions of system (1), i.e. solutions in the form Φ(t, x) = e iλ1t u(x), Ψ(t, x) = e iλ2t v(x), then system (1) is transformed into the following elliptic system where λ j > 0, β, γ are constants. A solution (u, v) is called nontrivial if u ≡ 0 and v ≡ 0; a solution (u, v) is semi-trivial if (u, v) is of type (u, 0) or (0, v). We call a nontrivial solution (u, v) positive if u > 0 and v > 0. When system (2) is only nonlinearly coupled, i.e. γ = 0, extensive research has been done regarding the existence, multiplicity and asymptotic behavior of nontrivial solutions to system (2). We refer to [4,7,11,13,14,22,28] and references therein. When system (2) is only linearly coupled, i.e. γ = 0, β = 0, Ambrosetti et al. [5] studied the existence and asymptotic behavior of the multi-bump solutions of system (2). When γβ = 0, to our best knowledge, only a few interesting results have been obtained in [9,24,33]. In particular, the second and third author of the current paper obtained some existence results for system (2) in [33] by using variational methods and bifurcation techniques.
Denote by w λj ,µj the non-degenerate positive radial solution of Note that, by the classical bootstrap argument, solutions of system (2) which are in X also in C 2 (R N ) × C 2 (R N ).
2. γ-bifurcation of fully symmetric system. Consider system (2) under symmetric assumptions Denote w = w 1,1 . It has been shown in [33] that problem (4) has a synchronized solution branch in R 2 × X, Let T w | β be the restriction of T w for fixed β, which is therefore only parameterized in γ. Bifurcations with respect to T w | β will be called γ-bifurcations. For fixed β ∈ (−1, +∞), finitely many γ-bifurcation points were found with respect to T w | β . Moreover, there is a global bifurcation branch emanating from each bifurcation point. In this section, we shall give more information about these global γ-bifurcation branches, where the word "global" is understood by taking (−1, 0]×X as the whole space. Precisely, we consider the following two interesting problems left in [33]: 1. How many γ-bifurcation points are there with respect to T w | β for a fixed β ∈ (−1, +∞)?
The bifurcation parameters depend on the following eigenvalue problem It is well known (see [6]) that problem (5) possesses a sequence of eigenvalues with η k → +∞ as k → +∞. Let

GUOWEI DAI, RUSHUN TIAN AND ZHITAO ZHANG
It is easy to see that f is a strictly decreasing function in (−1, +∞) and lim β→(−1) + f (β) = +∞. So we can denote β k := f −1 (η k ) for any k ≥ 1. In particular, we can see To save notations, we refer γ-bifurcation simply as bifurcation in the following theorem. Theorem 2.1. For any fixed β ∈ (−1, +∞), one has: there is a global bifurcation branch S β l ⊂ (−1, 0] × X of positive solutions which does not return to the synchronized branch, such that S β l satisfies one of the following three properties (i) meets infinity at γ = −1, (ii) meets infinity at γ = 0, Remark 1. An important difference lies between the γ-bifurcations and β-bifurcations is that the space dimension N now affects the behavior of synchronized solution branch T w . Precisely, using the Pohozaev identity, we have Figure 1 illustrates the possible global bifurcation branches at a bifurcation point (γ l , u l , v l ) ∈ T w , 1 ≤ l ≤ k. Each bifurcation branch either blows up at a γ = −1 or γ = 0, or reaches to {0} × X. The dashed lines represent the three possible situations of bifurcation branch and only one of them could happen.  It has been shown in [33] that system (2) is invariant under the following transformation: . Under this σ-invariance, one can easily get the symmetric bifurcation results about opposite sign solutions for γ ∈ [0, 1).
Hence η j (γ) is a continuous and strictly decreasing function of γ. For the right limit of η j at −1, we fix any E j−1 , then It has been shown in [33] that system (2) has no positive solution if λ 1 = λ 2 = 1, γ = −1 and β > 0. Here we improve this result by allowing β > −1. Proof. Assume for contradiction that (u, v) is a positive solution of system (4). Adding the two equations together, one sees that If β ∈ (−1, 1], On the other hand, if β > 1, Thus for any β > −1, So the Liouville type theorem [16, Theorem 2.3] implies U ≡ 0. This is a contradiction. The proof of Theorem 2.1 requires the following a priori bounds on positive solutions (γ, u, v), provided γ is bounded.
Proof. We follow a blow up procedure introduced by Gidas and Spruck [21]. Since the method is standard, we only sketch the argument. Assuming for contradiction that there is a sequence of solutions (γ n , u n , v n ) to system (4) such that Without loss of generality, we may assume that Then max Noting that γ n is bounded, passing to a subsequence if necessary, we see that (U n , V n ) → (U 0 , V 0 ) C 2 loc R N as n → +∞, which is a nontrivial and nonnegative bounded radial solution of It follows from [16, Theorem 2.1] that (U 0 , V 0 ) = (0, 0), which contradicts U 0 (0) = 1. Now we derive a uniform decaying rate of positive solutions to system (4) in X. Lemma 2.5. For any fixed β ∈ (−1, +∞), γ ∈ (−1, 0] and given positive solu- Let ζ = r (N −1)/2 U and denote by then it follows that Case 1. If Q remains non-positive in (R 1 , +∞), then by an argument similar to that of [32, Theorem 3], we have (e r ζ α ) ≤ 0 and Since U (r) → 0 as r → +∞, there exists R 3 ≥ R 2 such that U (r) ≤ 1 for any r ≥ R 3 . Thus Thus Case 2 does not occur. Therefore, there exists M > 0 such that U (r) ≤ M r (1−N )/2 e −r(1+γ)/2 for large enough r. Since u, v are nonnegative, the exponential decay of u and v follows.
To show the exponential decay of u and v , we first note that h(r) := e r ζ α is decreasing in (R 1 , +∞) since (e r ζ α ) ≤ 0 for r ≥ R 1 . It follows that is decreasing in (R 1 , +∞). Hence r N −1 U (r) ≤ 0 for any r ≥ R 1 . By (11), we find that This combing with the exponential decay of U implies that r N −1 U is increasing in which contradicts the exponential decay of U . Hence, l ∞ = 0. When N = 2, one has U ∈ L 1 (0, +∞), which contradicts U ∈ L ∞ (0, +∞). So, we still get l ∞ = 0.
Integrating (12) from r to +∞, It follows that for r large enough. The proof is completed. Proof. Suppose on the contrary that there exists a sequence where It is well known that u k and v k are monotone decreasing on [0, ∞) for β ≥ 0. On the other hand, if β ∈ (−1, 0), we cannot conclude the monotonicity of u k and v k as above, since the system is no longer cooperative. Whereas, we can employ the idea of "moving planes" method to show that u k and v k are monotone decreasing for r sufficiently large.
Claim. There exists λ * > 0 such that u k and v k are monotone decreasing in [λ * , +∞) for any k ∈ N.
loc R N , there exist r * > 0 and k * ∈ N such that q k (r) + p(r) ≥ (1 + γ) /2 for any r ≥ r * and k ≥ k * , where p(r) is the same as that of Lemma 2.5. Then as that of Lemma 2.5, we can show that there exists M > 0 which is independent on (u k , v k ) such that u k , v k ≤ M r − N −1 2 e −r(1+γ)/2 for r ≥ r * and k ≥ k * , which is a contradiction. Finally, by an argument similar to that of Lemma 2.5, we can obtain the exponential decay of |u k + v k |.
With Lemma 2.4 and Lemma 2.6 in hand, we obtain the following uniform boundedness of nonnegative solution of system (2) in X. Proof. Suppose on the contrary that there exists a nonnegative solution sequence {(γ n , u n , v n )} of system (4) with γ n ∈ [γ, γ], such that u n + v n → +∞ and γ n → γ * ∈ [γ, γ] as n → +∞. Note that when γ * = γ or γ * = γ, the limit should be understood as one-side limit.
From Lemma 2.6, we see that there exist R > 0 and M > 0 such that for any r ≥ R and any n ∈ N. Invoking Lemma 2.4, we have that It follows that u n L 2 is uniformly bounded. Similarly, we have that v n L 2 is uniformly bounded. So we must have ∇u n L 2 → +∞ or ∇v n L 2 → +∞ as n → +∞. Without loss of generality, we assume that ∇u n L 2 → +∞ as n → +∞.
Multiplying the equation for u in system (2) by u n / ∇u n 2 L 2 and integrating on R N , then using Lemma 2.4, we obtain that as n → +∞, which is a contradiction.
To guarantee the positiveness of bifurcation solutions, we consider the following modified system of (2) where u + = max{u, 0} and v + = max{v, 0}. For any γ ∈ (−1, 0) and β ∈ R, system (18) has only nonnegative solutions. To see this fact, let u − = min{u, 0} and v − = min{v, 0}. Then multiplying the first equation with u − , the second with v − and integrating, we obtain that where the facts γu − ≥ 0, v − ≤ v, γv − ≥ 0 and u − ≤ u in R N are used. Adding the above two inequalities together, we have that It follows that u − ≡ 0 and v − ≡ 0. Therefore, u ≥ 0 and v ≥ 0 in R N . Since (18) does not have semi-trivial solution, for any γ ∈ (−1, 0) and β ∈ R, the strong maximum principle implies that every nontrivial solution of system (18) where   G(λ, u), B r (0)) changes when λ passes µ, then S possesses a maximal subcontinuum C µ ⊂ O such that (µ, 0) ∈ C µ and one of the following three properties is satisfied by C µ : If O is bounded, Lemma 2.8 is just Corollary 1.12 of [29]. So Lemma 2.8 can be seen the complement of Corollary 1.12 of [29].
Proof of Theorem 2.1. The local bifurcations of system (4) has been established in [33], therefore we shall concentrate on the number of bifurcations with respect to T w | β and also the behavior of global bifurcations.
From now on, we fix β ∈ [β k+1 , β k ) and l ∈ {1, 2, ..., k}. Define the functional I + γ : X → R of (18) by Let (u γ , v γ ) be a critical point of I + γ on T w | β . By arguments similar to [6, Lemma 3.3] with obvious changes, we get where m(γ) is the index of the quadratic form D 2 I γ (u γ , v γ ). Since every eigenvalue of problem (6) has multiplicity one, so by (20), deg ∇I + γ , (u γ , v γ ) changes when γ crosses a value γ l . Now it follows from Lemma 2.8 by taking O = (−1, 0) × X that there exists a global γ-bifurcation branch S β l occurs at (γ l , u γ l , v γ l ). According to [33], the kernel space of linearized system of (4) is spaced by {(φ l , −φ l )}, where φ l is the eigenfunction corresponding to η l (γ l ) of (6). Note that φ l has l − 1 simple zeros. Now for (γ, u, v) ∈ S β l near the bifurcation point (γ l , u γ l , v γ l ), it follows from [29, Lemma 1.24] that That is to say, S β l is curve near the bifurcation point (γ l , u γ l , v γ l ) in space X and then also in C 1 R N × C 1 R N by bootstrap the perturbation term o(γ − γ l ). Therefore has precisely l − 1 simple zeroes provided γ is close to γ l . Let S j denote the set of functions in H 1 r R N also in C 1 loc R N by bootstrap argument which have exactly j − 1 interior simple zeroes for any j ∈ N. So u − v ∈ S l provided γ is close to γ l .
We claim that u − v ∈ S l for any (γ, u, v) ∈ S β l .
Suppose on the contrary that there exists (γ, u, v) ∈ S β l such that u−v ∈ S l . Then there exists (γ * , u * , v * ) ∈ S β l such that h := u * − v * has a double zero because S l is an open set (see [29]). Without loss of generality, we may assume that (γ * , u * , v * ) be the first such kind of point as γ moves away from γ l along S l . Then one can see that h satisfies the following equation Then by an argument similar to [6, Theorem 2.3], we can easily show that h(r) ≡ 0. Thus S β l returns to the synchronized branch at γ = γ * . Since (0, u 0 , v 0 ) is not a bifurcation point of system (2) with respect to T w | β , we have γ * = 0. It follows that γ * = γ m for some m ∈ {1, 2, . . . , k} \ {l}. Reasoning as the above, u − v has precisely m − 1 simple zeroes provided γ is close to γ m for any (γ, u, v) ∈ S β l . So there must exist a point (γ * , u * , v * ) ∈ S β l with γ * ∈ (γ l , γ m ) ∪ (γ m , γ l ) such that u * − v * has a double zero. Similarly, one has u * − v * ≡ 0. This contradicts the fact that (γ * , u * , v * ) be the first such kind of point.
So we obtain that u − v has precisely l − 1 simple zeroes for any (γ, u, v) ∈ S β l \ {(γ l , u γ l , v γ l )} and S β l does not return to the synchronized branch for any 1 ≤ l ≤ k. It follows that S β l ∩ (T w | β ) = (γ l , u γ l , v γ l ), i.e. the third alternative of Lemma 2.8 cannot occur. Moreover, by Lemma 2.3, no bifurcation branch can reach to the boundary {−1} × X. On the other hand, if S β l meets the boundary {0} × X, then the corresponding solution (u, v) cannot be semitrivial, due to the nodal property of u − v and u, v are nonnegative. At last, by Lemma 2.7, the remaining possibilities of Lemma 2.8 then give us part (a).
By an argument similar to that of [22,Corollary 2.4] with obvious changes, we have the following compactness result. is compact in X.
Now we describe the five dimensional bifurcation branches. In fact, any positive solutions of system (21) with β > 0 must be radially symmetric and decreasing (see [10]). To find positive solutions, we confine the problem to the nonnegative cone Then we shall find solutions in R 5 + × P via bifurcation technique. In order to state the main results, it is convenient to introduce some notations. Let be the set of trivial solutions and We also need the function ξ : R + → R + defined by ξ(s) := inf where w = w 1,1 is the unique positive ground state solution of (3). Without loss of generality, we may assume that µ 1 ≤ µ 2 .
The main result of this section is the following theorem.
If λ 1 = λ 2 and µ 1 = µ 2 , Bartsch-Wang-Wei also showed that Although the conditions λ 1 = λ 2 and µ 1 = µ 2 are not written out explicitly in [8, Theorem 1.1], they are used in their argument (see [8, P. 361]) in order to get . While, we do not need these conditions in Theorem 3.2. So Theorem 3.2 improves the corresponding ones of [8] in this sense.
The energy functional associated with system (21) is As a consequence of the Sobolev embedding theorem, E b : X → R is well-defined and is a C 2 -functional. The gradient of E b with respect to ·, · b can be computed as where Λ = diag (λ 1 , λ 2 ) and f b ( − → u ) = µ 1 u 3 + βuv 2 , µ 2 v 3 + βvu 2 . By the compact embedding of H 1 r R N → L q R N for 2 < q < 2 * the map is completely continuous. It has been shown that in [8] the only possible bifurcation point on T j is b j . Since P ⊂ X is closed and convex, there exists a retraction r : X ⊂ P. To find the fixed point of A b in P is equivalent to find − → u ∈ X such that Proof of Theorem 3.2.
Then F is a completely continuous mapping satisfying F (b, 0) = 0 whenever (b, 0) ∈ O. We refer to R 5 × {0} ∩ O as the trivial solutions. Clearly, Clearly, h is continuously differentiable and has 0 as a regular value. Let Γ = h −1 (0), then b * ∈ Γ. For any ε > 0, taking It is easy to see that b and b lie in the same component of Γ. Choosing ε small enough, we can assume that neither b nor b are Γ bifurcation points of F := I − F . Lemma 2.4 of [8] shows that Now, all the assumptions of Theorem 2.4 of [20] are satisfied. So there exists a closed connected subset, , whose dimension at each point is at least 5, and C 1 ∩ Γ = b * . Moreover, one of the following three properties is also satisfied by C 1 : [8] implies that the third alternate does not occur. Let Then C 1 bifurcates from T 1 at b * and satisfies the alternatives (a) or (b). From Lemma 3.1, we can easily get that if B ⊂ R 4 + × [0, +∞) is compact then there exists a uniform bound R > 0 such that Similarly, we can get the set C 2 emanating from T 2 at b 2 and satisfying the desired conclusions.
The difference between Corollary 2 and [6, Theorem 2.3] is the conclusion at β = 1. In [6], Bartsch-Dancer-Wang showed that at the point β = 1 the bifurcating solutions are explicitly given by (22) with λ = µ = 1 and there are no further solutions of problem (21) near (1, w, 0) or (1, 0, w). While, we can see from Corollary 2 that problem (21) with β = 1 only has positive solutions that are explicitly given by (22) with λ = µ = 1. So Corollary 2 improves the corresponding results of [6]. Proof of Theorem 4.2. Assume that (u, v) be a positive solution of system (21) at β = µ. Then we multiply the first equation in problem (21) by u, the second equation by u 2 / (v + ε) for any ε > 0 small enough, and integrate resulting equations over R N . This yields In particular, we take ε = 1/n. Then we have that Γ 1/n ≤ µu 4 + µu 2 v 2 := F.
Since the embedding of H 1 r R N → L 4 R N is compact, one has that So F is a integrable function defined on R N . On applying the Dominated Convergence Theorem we find that lim n→+∞ R N Γ 1/n (x) dx = R N lim n→+∞ Γ 1/n (x) dx = 0.
We first consider the case of µ 1 < µ 2 . In this case, it is well known that problem (21) does not have a nontrivial solution with nonnegative component if β ∈ [µ 1 , µ 2 ]. So S 1 cannot go to right in the direction of β. From Lemma 3.1, we can see that C 1 cannot blow up on any bounded set of β. It follows that S 1 cannot blow up on any bounded set of β. So S 1 must meet {0} × X. Similarly, S 2 cannot go to left in the direction of β, so it must coincide with {(β, u 0 , v 0 ) : β > µ 2 }.
We now consider the case of µ 1 = µ 2 . From Theorem 4.2, we have that (21) has infinitely many positive solutions at β = µ which must be the form of (22). So S 1 still cannot go to right in the direction of β. Then reasoning as above, we can obtain the desired conclusions.