Abstract
In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball. In addition, the nonexistence of the ground state solutions under certain conditions on the nonlinearities and the complete structure of different types of solutions to the shooting problem are proved.
Introduction
In this article, we study the solutions of the nonlinear elliptic system (1.1) can be viewed as a high-dimensional counterpart of a well-known equation:
u − u + u p = 0, in R n , u(x) → 0, |x| → ∞, (1.2) where p > 0. The uniqueness of positive solutions of (1.2) for p ∈ (1, (n + 2)/(n − 2)) was first proved in [20] , and simplifications of proof and generalizations can be found in, for example, [3, 6, 21, 29, 30] .
To cite a source of (1.2), we consider the following stationary Keller-Segel system, which is related to a model to describe the chemotactic aggregation stage of cellular slime mold (see, e.g., [11, 24] ): 
v(x),
(KS * ) can be reduced to (1.2) on a bounded domain.
Another motivation for studying the scalar equation related to (1.1) arises from the nonlinear Schrödinger systems (see, e.g., [1, 4, 15, 25] )
where λ 1 , λ 2 , μ 1 , μ 2 > 0 and β ∈ R. If we treat specific solutions (u 0 
where B R (0) ⊂ R n is the ball centered at the origin with radius R, has been investigated by many mathematicians. The existence of solutions of (LE) or more general type was established in [10, 12, 13, 18, 31, 35] for p, q 1, pq = 1, n 3 and satisfying the subcritical condition
On the other hand, if p, q > 0, n 3 and are supercritical, i.e.,
the nonexistence of solutions of (LE) was obtained in [27, 37] . Moreover in [13, 19] , the uniqueness of positive radial solutions to (LE) has been derived for p, q > 0 with pq = 1 as well. We remark that, in [12, 13, 19] , the scaling argument is a key to the proof of the uniqueness due to the homogeneity of the nonlinearities in (LE). The existence of positive solutions to (1.1) for subcritical p, q was considered in [16] , but the uniqueness of solutions to (1.1) is still not known. For the nonlinear Schrödinger systems (NS), a few uniqueness results were obtained recently for some special cases [23, 26] .
In this paper, we apply linearization techniques and the implicit function theorem to prove the uniqueness of positive solutions to (1.1). While such techniques have been widely used in proving the uniqueness of solutions of scalar equations (see [6, 20, 21, 29, 30] ), the generalization to systems of equations is not straightforward due to the coupling. Here we use some ideas appearing in our earlier works [7, 8] .
By virtue of the method of moving planes, the Dirichlet problem of (1.1) in B R (0) with (u, v) = (0, 0) on ∂ B R (0) can be reduced to a corresponding system of ordinary differential equations with u(R) = v(R) = 0, see for example [2, 5, 8, 14, 22, 36] . While the proof is standard, for the reader's convenience, we will present it in detail in Appendix A.
Throughout this paper, we investigate the radial solutions of (1.1) by considering the following initial value problem (here we extend the definition of the nonlinearities so solutions can be defined for u, v 0): 
Furthermore, the corresponding initial data of such solutions at the origin, denoted by α 1 (R) and α 2 (R), are increasing in R > 0.
In addition to the existence and uniqueness of solutions to the Dirichlet problem for (1.1) stated above, we also present the nonexistence of the ground state solutions and describe a complete structure of solutions of (1.3) in terms of initial data under certain conditions on the nonlinearities. The results in this part require a stronger condition on p and q: 0 < p < 1 and 0 < q < 1 instead of 0 < pq < 1. Before we introduce other main results, various types of solutions for (1.3) are introduced as follows.
3) is classified into the following types:
We use the following notations for the regions of initial data corresponding to various types of solutions for (1.3):
Our next result is the nonexistence of ground state solutions of (1.3). With no Type G solution of (1.3), the following result completely characterizes the structure of initial data sets corresponding to solutions of remaining Types B, C, C u and C v . such that
Geometrically the set Ω C ∪ T ∪ Ω B separates the two open subsets S u and S v , see Fig. 1 for the illustration of the structure of Ω C , Ω B , S u and S v . While the set Ω C is proved to be a curve, it is still unclear whether Ω B is also a curve. Most results in Theorem 1.3 still hold if we replace the assumption 0 < p < 1, 0 < q < 1 by merely p, q > 0 and 0 < pq < 1, see Remark 3.1 for details.
It is possible to obtain an even refiner description of S u and S v . For that purpose, we define Accordingly we also define
Then the following results hold for the further partitions of S u and S v (see Fig. 1 
We believe that our method in this paper will be useful for some other more general nonlinear elliptic systems on bounded or unbounded domains of general dimensions. We also mention that result like Theorem 1.1 can be proved using other method even for a general bounded domain, see [9] , but our main emphasis here is to present a systematic approach for radially symmetric solutions and obtain a complete structure of solutions to the shooting problem (1.3) , that is information not provided in [9] . We also mention that the shooting problem like (1.3) can be solved numerically, and bifurcation diagrams showing the regions of different types of solutions as in Theorems 1.3 and 1.4 can be numerically obtained [17] .
This article is organized as follows. In Section 2, some preliminaries are prepared. In Section 3, we give proofs of Theorems 1.2, 1.3 and 1.4. By investigating the corresponding linearized systems and employing the implicit function theorem, a complete demonstration of Theorem 1.1 will be offered in Section 4. Finally, Section 5 is devoted to some generalizations which can be done by similar methods, and after that, a detailed proof of the symmetry of solutions to the Dirichlet problem is provided in Appendix A.
Preliminaries
First we show the global existence of the solutions of (1.3). 
Then the maximum principle implies that
Hence u(r) cannot blow up to +∞ at r = R. Similarly, v(r) cannot blow up to +∞ at r = R as well.
If u(r) and v(r) vanish at some points (not necessary to be the same), then from Remark 1.1, we see 
The following lemma gives the monotone properties of ϕ i and ψ i (i = 1, 2).
for some c 1 > 0, which implies
Similarly we have
for some c 2 > 0, and hence
Since the inequalities in (2.4) and (2.5) hold whenever ϕ 1 (r) > 0 and ψ 1 (r) < 0, we deduce that 
We assume that u(r) > 0, v(r) > 0 for all r > 0, and it does not belong to Type G, Type B, or (1, 1). We claim that there exists r 0 0 such that
Proof of Claim. We prove it for all possible (α 1 , α 2 ). 
That is a contradiction since (1, 1) is the only point in the first quadrant so that u − v p = 0 and v − u q = 0.
Hence either u (r) or v (r) must change sign. We define 
for some C > 0. This yields a contradiction since v(r) is positive on [0, ∞). We complete the proof of this lemma. 2
From the proof of Lemma 2.3, the regions S u and S v are nonempty.
Corollary 2.1. Let S u and S v be defined in (1.5) . Then:
Remark 2.1. By the arguments described in the proof of Lemma 2.3, we see that if pq < 1 and
Nonexistence and structure of solutions
For the proof of Theorem 1.2, the following lemma is a key step. 
for all r > 0, and consequently 0 < u(r) < α 1 < 1 and 0 < v(r) < α 2 < 1 for all r > 0.
We prove the lemma by contradiction. Suppose (3.1) is not true, then 
Letting r 1 → ∞ in the above inequality, we get
where Without loss of generality, we assume that q p.
q for large r, 
for any k ∈ N and some C > 0. This is impossible because q < 1 which deduces
This completes the proof of Theorem 1.2. 2
In order to prove Theorems 1.3 and 1.4, we derive a geometric property of the regions defined in (1.5) which is an essential element to clarify the structure of solutions on (α 1 , α 2 )-plane.
, and
and (α * 1 , α 2 ) ∈ S v any for 0 < α 2 < α *
.
Proof. We only prove the results involving S u . The others involving S v are similar, and we omit the details.
and let
where 
The functions (φ, ψ) defined in (3.6) and (3.7) are also defined on (0, ∞), and φ(r) > 0 and ψ(r) < 0 for r ∈ (0, ∞). Since u(r; α * 1 , α * 2 ) → 0 and
Then φ(x) must achieve its positive maximum value at x 0 ∈ R n and φ(x 0 ) 0, which contradicts with φ − φ 0 in R n . Hence R = ∞ is impossible and R is finite, then similar to the proof above,
Then from Corollary 2.1, Proposition 3.1 and Theorem 1.2, we obtain that
which is a nonempty simply connected closed subset of R 2 + , and 
From Proposition 3.1, it is easy to see that γ 1 = γ 2 which is strictly increasing on (0, 1). So results in Theorem 1.3 hold if we define γ ( 
and parts (b) and (c) in Theorem 1.3 still hold. On the other hand, Proposition 3.1 implies that part (a) in Theorem 1.3 now becomes
Theorem 1.2 excludes the possibility of ground states when p < 1, q < 1, and it is unclear whether the ground state exists when only pq < 1 holds.
Existence and uniqueness
This section is devoted to the existence and uniqueness of solutions to the Dirichlet problem (1.1) in the ball with zero boundary condition: To attain our uniqueness result, we introduce the following functions: 
for r > 0. The following assertions play a significant role in proving the uniqueness of solutions of (4.2). Lemma 4.1. Let p, q > 0 and pq < 1. If (u(r), v(r)) is a solution of (4.2), then C Φ (r) is strictly decreasing and  C Ψ (r) is strictly increasing on (0, R] . Furthermore, C Φ (R) > C Ψ (R).
Proof. First, by Remark 4.1(i), we have C Φ (r) > C Ψ (r) for r ∈ (0, r 0 ) for some 0 < r 0 R. We claim that C Φ (r) < 0 and C Ψ (r) > 0 for r ∈ (0, r 0 ). not strictly decreasing on (0, r 0 ) . Then by Remark 4.1(ii), there exist 0 < r 1 < r 2 r 0 such that
Proof of Claim. Suppose that C Φ (r) is
By combining (4.3), (4.4) and Lemma 2.2, we obtain 6) which implies that Φ(r; C 0 ) has a local minimum at somer ∈ (r 1 , r 2 ) and Φ (r; C 0 ) 0. However, from (4.5) and (4.6), we have
This is a contradiction. The proof for C Ψ (r) is similar and we complete the proof of this claim. 2
Now, suppose there exists
Moreover, from (1.3) and (4.5), we get
Hence, by integrating the above equality from 0 to R 0 , we obtain
Similarly, we also have
(4.9)
Since pq < 1, (1 − pq)u q (r) > 0. Therefore from (4.7), (4.8) and (4.9), we deduce
which is impossible. Hence the graphs of C Φ and C Ψ do not intersect on [0, R]. The proof of this lemma is complete. 2
Based on Lemma 4.1, it is easy to obtain the following consequences.
Lemma 4.2. Let p, q > 0 and pq < 1. If (u(r), v(r)) is a solution of (4.2) and define
,
then Φ(r; C ) and Ψ (r; C ) satisfy the following properties.
Now we prove Theorem 1.1.
Proof of Theorem 1.1. We split the proof into several steps.
Step 1. (Non-degeneracy) We show that each solution of (4.2) is non-degenerate, hence the solution set of (4.2) is locally a smooth curve. Let (u(r; α 1 , α 2 ), v(r; α 1 , α 2 )) be a solution of (4.2). We define
, r > 0, (4.10) and 
is the initial data corresponding to solutions of (4.2) for R ∈ (R 0 − ε, R 0 + ε). Next, we claim β 1 (R) > 0 and β 2 (R) > 0 for R ∈ (R 0 − ε, R 0 + ε). By (1.3) and Lemma 3.1, we get
(4.13)
By combining (4.11) and (4.13), we assure that β 1 (R) and β 2 (R) have the same sign. Moreover, (4.11) and (4.12) also imply 
This proves that each connected component of the solution set of (4.1) is a curve which can be parameterized by R.
Step 2. (Existence) We show the existence result by applying the monotone iteration method in [32, 33] . First, let
Then it is easy to see that (u * (x), v * (x)) is a super-solution of (4.1). Since pq < 1, then without loss of generality, we can assume that 0 < p q and in particular 0 < p < 1. For a sub-solution, we define
where φ 1 > 0 is the eigenfunction corresponding to the principal eigenvalue λ 1 > 0 of (4.14) and ε 1 , ε 2 satisfy ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 
) is a sub-solution of (4.1). By choosing ε 1 and ε 2 sufficiently small such that
and applying the extension of monotone methods to systems in [33] , we prove the existence of a positive solution of (4.1) for any R > 0.
Step 3. (Uniqueness) From Step 2, for any fixed R * > 0, there exists a positive solution (u(x; R * ),
This curve can be extended to a maximum curve Γ in R 3 + by repeatedly applying Step 1. (R 2 ,β 1 ,β 2 ) still represents a positive solution of (4.1) so Step 1 can be applied to extend Γ further right, which contradicts with the maximality of Γ . Therefore R 2 = ∞ and Γ = {(R,
Suppose that (4.1) has positive solutions other than the ones on Γ , then using the exact same proof, one can show that such solutions are on another branchγ = {(R,β 1 (R),β 2 (R)): R ∈ (0, ∞)} satisfying β i (R) > 0 and lim R→0 +β i (R) = 0, i = 1, 2. Due to the monotonicity, both γ andγ can be re-parameterized by α 1 (the initial value of u). [32, 33] ) is well known to be used to prove the existence of solutions, the technique applied to the proof of Theorem 1.1 can provide us with a new way to construct a desired sub-solution or super-solution involving the power of nonlinearities for more general systems of PDEs.
Remark 4.3.
If we can show that Ω G = ∅ for the case considered in Theorem 1.1, then lim R→∞ β i (R) = β i = 1 in the proof of Theorem 1.1 (that is the case if 0 < p < 1 and 0 < q < 1). Otherwise (β 1 ,β 2 ) could be an element in Ω G ifβ i < 1 for at least one of i = 1, 2.
Generalizations
The methods presented above can be used to prove existence, uniqueness of solutions in other similar situations. As an example other than (1.1), we consider
where f 1 and g 1 are non-negative, unbounded C 1 functions on R + which satisfy
and we also extend f 1 and g 1 to C (R) so that
We investigate the initial value problem, i.e., the radial case of (5.1): We call the solution stated in Theorem 5.1 the Dirichlet-type solution. In addition, we can also clarify the structure of all solutions to (5.4). Similar to before, we call the solution (u, v) of (5.4) is a ground state if (u(r), v(r)) → (0, 0) as r → ∞. Also, a u-crossing (resp., v-crossing) solution (u, v) is that u (resp., v) vanishes at some finite point where v (resp., u) is still positive. Hence, by the above inequality and (A.4), we conclude x 0 ∈ ∂Σ η u ∩ T η u and, by (A.6),

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