Abstract
In this article, we mainly construct multiple blowingup and concentrating solutions for a class of Liouvilletype equations under mixed boundary conditions:
for ε small, where , Ω is a bounded, smooth domain in , Γ := {p_{1}, ..., p_{N}} ⊂ Ω is the set of singular sources, δ_{p }denotes the Dirac mass at p, ν denotes unit outward normal vector to ∂Ω and b(x) > 0 is a smooth function on ∂Ω.
2000 Mathematics Subject Classification: 35B25; 35J25; 35B38.
Keywords:
multiple blowingup and concentrating solution; Liouvilletype equation; singular source; mixed boundary conditions; finite dimensional reduction1 Introduction
In this article, we mainly investigate the mixed boundary value problem:
for ε small, where , Ω is a bounded, smooth domain in , Γ:= {p_{1}, ..., p_{N}} ⊂ Ω is the set of singular sources, δ_{p }denotes the Dirac mass at p, ν denotes unit outward normal vector to ∂Ω and b(x) > 0 is a smooth function on ∂Ω.
Such problems occur in conformal geometry [1], statistical mechanics [24], ChernSimons vortex theory [511] and several other fields of applied mathematics [1216]. In all these contexts, an interesting point is how to construct solutions which exactly "blowup" and "concentrate" at some given points, whose location carries relevant information about the potentially geometrical or physical properties of the problem. However, the authors mainly consider the Dirichlet boundary value problem, and little is known for the problem with singular sources satisfying α_{i }∈ (1, 0) for some i = 1, ..., N. The main purpose of this article is to study how to construct multiple blowingup and concentrating solutions of the Equation (1) with the mixed boundary conditions and singular sources.
Let G_{t,ε }denotes the Green's function of ∆ with mixed boundary conditions on Ω, namely for any y ∈ Ω,
and let H_{t,ε}(x, y) = G_{t,ε}(x, y) + log x  y be its regular part. Set G_{1 }= G_{1,ε }and H_{1 }= H_{1,ε}. Since ε exactly disappears in the Equation (2)_{t = 1}, G_{1 }and H_{1 }don't depend on ε. The Equation (1) is equivalent to solving for , the regular part of v, the equation
Thus, we consider the more general model problem:
where is a smooth function such that f(pi) > 0 for any i = 1, ..., N. Set Ω' = {x ∈ Ω: f(x) > 0}, and Δ_{m }= {p = (p_{1}, ..., p_{m}) ∈ Ω^{m}: p_{i }= pj for some i ≠ j}.
It is known that for , or α_{i }= 0 for any i = 1, ..., N, if u_{ε }is a family of solutions of the Equation (3)_{t = 1 }with inf_{Ω }f > 0, which is not uniformly bounded from above for ε small, then u_{ε }blows up at different points with n + m ≥ 1, 0 ≤ n ≤ N, and , and satisfies the concentration property:
in the sense of measures in . Moreover, is a critical point of the function:
(see [7,1723]). An obvious problem for the Equation (3) is the reciprocal, namely the existence of multiple blowingup solutions with concentration points near critical points of φ_{n,m}.
The earlier result concerning the existence of multiple blowingup and concentrating solutions of the Equation (3) is given by Baraket and Pacard in [24]. When t = 1 and α_{i }= 0 for any i = 1,2, ..., N, they prove that any nondegenerate critical point of the function φ_{n,m }with n = 0 generates a family of the solutions u_{ε }which blowup at , and concentrate in the sense that (4) holds. Esposito [20] performs a similar asymptotic analysis and extends the previous result by allowing the presence of singular sources in the Equation (3)_{t = 1}, that is, . However, the asymptotic analysis method depends on the nondegenerate assumption of critical point of the function φ_{n,m }so much that it pays in return at a price of the very complicated and accurate control on the asymptotics of the solutions.
In fact, the finite dimensional reduction method, used successfully in higher dimensional nonlinear elliptic equation involving critical Sobolev exponent (see [6,25]), can avoid the technical difficulty in carrying out the asymptotic analysis method for the Equation (3). It is necessary to point out that the key step of the finite dimensional reduction is the analysis of the bounded invertibility of the corresponding linearized operator L of the Equation (3) at the suitable approximate solution. In [26,27], the authors construct the approximate solution, carry out the finite dimensional reduction and use some stability assumptions of critical points of φ_{0,m }to get the existence of multiple blowingup and concentrating solutions for the Equation (3)_{t = 1 }with , namely α_{i }= 0 for any i = 1,2, ..., N. When , a similar result for the Equation (3)_{t = 1 }under C^{0}stable assumption of critical point of φ_{n,m }(see Definition 4.1) is also established in [28].
Here in the spirit of the finite dimensional reduction, we try to extend the result of the Equation (1) in [20,28] by allowing the presence of singular sources with some α_{i }∈ (1, 0) and Robin boundary conditions with t ∈ (0,1). When we carry out the finite dimensional reduction, we need to get the invertibility of the desired linearized operator L for the Equation (3) under some α_{i }∈ (1, 0). Obviously, the linearized operator L easily produces the singularities at some singular sources with α_{i }∈ (1, 0), which makes trouble for the analysis of the bounded invertibility of L. But we can successfully get rid of it by introducing a suitable L^{∞}weighted norm (see (30) below) related with a "gap interval" (1, α_{0}), where α_{0 }= min{0, α_{1}, ..., α_{N}}. On the other hand, the presence of the term in the Equation (3)_{0<t<1 }brings some new technical difficulties. A flexible approach exactly helps us overcome the difficulties by making use of the maximum principle. In addition, a weaker stable assumption of critical points of the function φ_{n,m }also helps us construct multiple blowingup and concentrating solutions of the Equation (3). As a consequence, we have the following result.
Theorem 1.1 Let 0 ≤ n ≤ N and such that n + m ≥ 1. Assume that and is a C^{0}stable critical point for φ_{n,m }in (Ω' \ Γ)^{m }\ Δ_{m }with m ≥ 1 (see Definition 4.1). Then there exists a family of solutions u_{ε }for the Equation (3) with the concentration property (4), which blow up at ndifferent points in Γ, and mpoints in (Ω' \ Γ)^{m }\ Δ_{m }with φ_{n,m}(p*) = φ_{n,m}(p). Moreover, u_{ε }remains uniformly bounded on , and for any λ > 0.
Let us point out that from the proof of Theorem 1.1 Robin boundary condition can be considered as a perturbation of Dirichlet boundary condition for the problem (3) in using perturbation techniques to construct multiple blowingup and concentrating solutions. Based on this point, we also consider the DirichletRobin boundary value problem:
where T ⊆ ∂Ω is a relatively closed subset and . This together with other similar mixed boundary value problems can be founded in [29,30]. For the problem (6), we obtain the following result.
Theorem 1.2 Under the assumption of Theorem 1.1, then there exists a family of solutions u_{ε }for the Equation (6) with the concentration property (4), which blow up at ndifferent points in Γ, and mpoints in (Ω' \ Γ)^{m }\ Δ_{m }with φ_{n,m}(p*) = φ_{n,m}(p). Moreover, u_{ε }remains uniformly bounded on , and for any λ > 0.
Finally, it is very interesting to mention that to prove the above results we need to choose the classification solutions of the following Liouvilletype equation to construct concentrating solutions of the Equation (1) or (3):
given by
with if , c = 0 if (see [5,11,31,32]). Using these classification solutions scaled up and projected to satisfy the mixed boundary conditions up to a right order, the initial approximate solutions can be built up. Then through the finite dimensional reduction procedure and the notions of stability of critical points of the asymptotic reductional functional φ_{n,m}, multiple blowingup and concentrating solutions can be constructed as a small additive perturbation of the initial approximations.
This article is organized as follows. In Section 2, we will construct the approximate solution and rewrite the Equation (3) in terms of a linearized operator L. In Section 3, we give the invertibility of the linearized operator L, carry out the finite dimensional reduction and get the asymptotical expansion of the functional of the Equation (3) with respect to the suitable approximate solution. In Section 4, we give the proofs of Theorems 1.1 and 1.2.
2 Construction of the approximate solution
In this section, we will construct the approximate solution for the Equation (3). Let μ_{i}, i = 1, ..., N + m, be positive numbers and set
and
Obviously, Q_{i}(x) = S(x) for any i = N + 1, ..., N + m. Then the function
satisfies
Set {k_{1}, ..., k_{n}} ⊂ {1, ..., N} and k_{n+i }= N + i for any i = 1, ..., m.
We hope to take as an initial approximate solution of the problem (3). So we modify it to be
Lemma 2.1 For t ∈ (0, 1] and ,
uniformly in and in for ε small.
Proof. Set . Then z_{t}(x) satisfies
where
For any t ∈ (0, 1], it is easy to check . If t = 1, from the maximum principle and smooth function b(x) > 0, it follows
If 0 < t < 1, from the maximum principle with the Robin boundary condition (see [[33], Lemma 2.6]), it also follows
Thus using the interior estimate of derivative of harmonic function (see [[34], Theorem 2.10]), there holds
for any compact subset K of Ω, any t ∈ (0, 1] and any multiindex α with α ≤ 2, which derives (14) uniformly in and in for ε small. □
From this lemma we can construct the approximate solution , which satisfies the mixed boundary conditions. On the other hand, we hope that the error is smaller near every . In fact, we can realize this point by further choosing positive number such that
Consider the scaling of the solution of the Equation (3)
then v(y) satisfies
where . We also set and define the new approximation in expanded variables as V(y) = U(εy) + 4 log ε. Furthermore, set
and
Obviously, for all i = 1, ..., m.
Here, we want to see how well ∆V(y) match with W(y) through V(y). A simple computation shows
Then given a small number δ > 0, if for all i = 1, ..., n + m,
On the other hand, if for all i = 1, ..., n + m, obviously,
Now from (14), (15) and (17), we have
In summary, we set
and if for all i = 1, ..., n + m,
In the rest of this article, we try to find a solution v of the form v = V + ϕ of the Equation (16). In terms of ϕ, the problem (3) becomes
where
3 The finite dimensional reduction
In this section, we will carry out the finite dimensional reduction to solve the Equation (26). First of all, we need to get the desired invertibility of linearized operation L. Set
A basic fact to get the needed invertibility is that the linearized operator L formally approaches to the operator L_{i }under suitable dilations and translations, which have some wellknown properties that any bounded solution of L_{t}ϕ = 0 is
 a linear combination of z_{i0 }and z_{ij }for i = n + 1, ..., n + m, j = 1, 2 (see [24,35]);
 proportional to z_{i0 }for and i = 1, 2, ..., n (see [20,28,36]).
Remark 3.1 These properties of the operator L_{i }have been discussed in the above papers only if for i = 1, ..., n, or for i = n + 1, ..., n + m. In fact, if for some i = 1, ..., n, the operator L_{i }has also the corresponding properties.
Lemma 3.2 For , any bounded solution ϕ of
Proof. If we express the bounded solution ϕ of the Equation (28) in Fourier expansion form as follow
u_{n}(r) is a bounded nontrivial solution of the equation
Since any solution of ∆u = e^{u }in is given by the Liouville formula
for any meromorphic function F defined on , the function
with any and , is the solution of ∆u = z^{2α}e^{u }in . Moreover, its derivative with respect to a at a = 0
is a solution of the Equation (29) with r = z.
For n ≥ 1, since {ϕ_{n}(r), ϕ_{n}(r)} is a set of linearly independent solutions of the second order linear homogeneous ODE (29), any bounded solution is a linear combination of ϕ_{n}(r) and ϕ_{n}(r). However, ϕ_{n}(r) ( resp. ϕ_{n}(r) ) tends to 0 ( resp. ∞ ) as r ↦ 0 and ϕ_{n}(r) ( resp. ϕ_{n}(r) ) tends to ∞ ( resp. 0 ) as r ↦ + ∞, which implies that the Equation (29) _{n≥1} has no bounded nontrivial solution.
For n = 0, is a bounded solution of the Equation (29)_{n = 0}, that is, of the Equation (28). We claim that there does not exist the second linearly independent bounded solution of the Equation (29)_{n = 0}. Otherwise, let ω be another linearly independent bounded solution of (29)_{n = 0}. Writing ω(r) = c(r)ϕ_{0}(r), we get that
Then there exists a constant C > 0 such that
Hence, ω(r) ~ C log r for r small, which implies ω(r) is unbounded on (0, + ∞). It contradicts the assumption that ω is bounded. □
Let us denote
where χ(r) is a smooth, nonincreasing cutoff function such that for a large but fixed number R_{0 }> 0, χ(r) = 1 if r ≤ R_{0}, and χ(r) = 0 if r ≥ R_{0 }+ 1. Additionally, set α_{0 }= min{0, α_{1}, ..., α_{N}}. For any α ∈ (1, α_{0}), we introduce the Banach space
with the norm
Now to get the invertibility of the linearized operator L, we only need to solve the following linear problems: given h of class with β ∈ (0,1), for m ≥ 1 and 0 ≤ n ≤ N, we find a function ϕ and scalars c_{ij}, i = n + 1, ..., n + m, j = 1, 2, such that
and for m = 0 and 1 ≤ n ≤ N, we find a function ϕ such that
Proposition 3.1 (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε_{0 }and C such that for any points , l = n + 1, ..., n + m, in Ω', with
there is a unique solution , of the Equation (31), which satisfies
for all ε < ε_{0 }and t ∈ (0, 1]. Moreover, the map p' ↦ ϕ is C^{1 }and
(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε_{0 }and C such that there is a unique solution ϕ ∈ L^{∞}(Ω_{ε}) of the Equation (32), which satisfies
for all ε < ε_{0 }and t ∈ (0, 1].
These results can be established through some technical lemmas. First for the linear Equation (32) under the additional orthogonality conditions with respect to Z_{i0}, i = 1, ..., n + m, and Z_{ij}, i = n + 1, ..., n + m, j = 1, 2, we prove the following priori estimates.
Lemma 3.3 (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε_{0 }and C such that for any points , l = n + 1, ..., n + m, in Ω', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
one has
for all ε < ε_{0}.
(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε_{0 }and C such that for any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
one has
for all ε < ε_{0}.
Remark 3.4 The idea behind these estimates partly comes from observing the linear Equation (32) with h = 0 on bounded set for ε small. After a translation and a rotation so that Ω_{ε }converges to the whole plan , the Equation (32) approaches L_{i}ϕ = 0 in . As a result, the solution of the Equation (32) under the additional orthogonality conditions (37) should be zero.
Proof. Case (i): First consider the "inner norm" and the "boundary norm" , we claim that there is a constant C > 0 such that if Lϕ = h in Ω_{ε}, then
We will establish it with the help of suitable barrier.
Consider that the function is a radial solution in of
we define a bounded comparison function
with a > 0. Set . While for all i = 1, ..., n + m,
Moreover, according to (21) and (22), on the same region,
So if a is small enough to satisfy (1 + α)^{2}a^{2(1+α) }> C + 1, R_{a }is sufficiently large. As a result, by (42) and (43), for any R ≥ R_{a}, we have Z(y) > 0 and L(Z) < 0 in .
Let M be a large number such that for all i = 1, ..., n + m, . Consider now the solution of the problem
A direct computation shows
where
and
For the sake of the convenience, we choose R larger if necessary. Then it easily see that these functions ψ_{i}, i = 1, ..., n + m, have a uniform bound independent of ε.
Now we can construct the needed barrier:
It is easy to check that in , and on . Since Z(y) > 0 and LZ(y) < 0 in , from the maximum principle (see [[37], Theorem 10, Chap. 2 ]), it follows that in . Similarly, in , which derives the estimate (41).
We prove the priori estimate (38) by contradiction. Assume that there exist a sequence ε_{k }→ 0, points , l = n + 1, ..., n + m, in Ω' which satisfy relation (33), functions h_{k }with ║h_{k}║_{n,m }→ 0, solutions ϕ_{k }with ║ϕ_{k}║_{∞ }= 1, such that
Then from the estimate (41), ║ϕ_{k}║_{l }≥ κ or ║ϕ_{k}║_{o }≥ κ for some κ > 0. Briefly set ε:= ε_{k}, . If ║ϕ_{k}║_{l }≥ κ, with no loss of generality, we assume that for some i Then if we set and satisfies
for z ∈ B_{R}(0). Obviously, for any we easily get in L^{q}(B_{R}(0)). Since is bounded in L^{q}(B_{R}(0)) and , elliptic regularity theory readily implies that converges uniformly over compact subsets near the origin to a bounded nontrivial solution of the equation
From Lemma 3.2, this equation implies that is proportional to z_{i0 }for i = 1, ..., n, or a linear combination of z_{i0 }and z_{ij }for i = n + 1, ..., n + m, j = 1, 2. However, our assumed orthogonality conditions (37) on ϕ_{k }pass to limit and yield the corresponding conditions (37) on , which means . Hence, it is absurd because is nontrivial.
If ║ϕ_{k}║_{o }≥ κ and ║ϕ_{k}║_{l }→ 0, there exists a point q ∈ ∂Ω and a number R_{1 }> 0 such that with . Consider and let us translate and rotate Ω_{ε }so that q' = 0 and Ω_{ε }approaches the upper halfplan . Since for all i = 1, ..., n + m, satisfies
with . Moreover, we easily get h_{k}(y) → 0 in . While t = 1, it is obvious to see that on ∂Ω_{ε}. So it is absurd because of . On the other hand, for any t ∈ (0,1), elliptic regularity theory with the Robin boundary condition (see [30,34,38] and the references therein) implies that converges uniformly on compact subsets near the origin to a bounded nontrivial solution of the equation
with . It follows that its bounded solution is zero. Hence, it is also absurd because is nontrivial, which derives the priori estimate (38) of the case (i). Since the proof of the case (ii) is similar to that of the case (i), we omit it.□
We will give next the priori estimate for the solution of the Equation (32) that satisfies orthogonality conditions with respect to Z_{ij}, i = n + 1, ..., n + m, j = 1, 2, only.
Lemma 3.5. (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε_{0 }and C such that for any points , l = n + 1, ..., n + m, in Ω', which satisfy the relation (33), and any solution ϕ of the Equation (32) with t ∈ (0, 1] under the orthogonality conditions
one has
for all ε < ε_{0}.
(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε_{0 }and C such that for any solution ϕ of the Equation (32) with t ∈ (0, 1], one has
for all ε < ε_{0}.
Proof. Case (i): Let ϕ satisfy the Equation (32) under the orthogonality conditions (44). We will modify ϕ to satisfy the orthogonality conditions (37). To realize this point, we consider some related modifications with compact support of the functions Z_{i0}, i = 1, ..., n + m.
Let R > R_{0 }+ 1 be large and fixed, and let be the solution of the equation
A simple computation shows that this solution is explicitly given by
Set
where η_{1}(r) and η_{2}(r) are smooth cutoff functions with the properties: η_{1}(r) = 1 for r < R, η_{1}(r) = 0 for ; η_{2}(r) = 1 for r < δ, η_{2}(r) = 0 for . We define a test function
Obviously, if , and if , in particular, near ∂Ω_{ɛ}.
Now we modify ϕ to satisfy the orthogonality conditions with respect to Z_{i0}, i = 1, ..., n+ m, and set
where the numbers d_{i }are chosen such that
Thus
and satisfies all the orthogonality conditions in (37). From Lemma 3.3 (i), we have
In order to get the estimate (45) of ϕ, we need to give the sizes of d_{i }and for any t ∈ (0, 1]. Multiplying the first equation of (47) by , integrating by parts and using the mixed boundary conditions of (47), we get
where . A simple computation shows that , which in combination with (48) and (49) yields
From some similar computations (see [[26], Lemma 3.2] and [[28], Lemma 3.3]), there exists a constant C > 0 independent of ε such that
and
which combined with (50) yields
Furthermore, from (48), (51), (53), and the definitions of and , we have
Similarly, the proof of the case (ii) can also be done, we omit it. □
Proof of Proposition 3.1. Case (i): From Lemma 3.5 (i), and the Fredholm's alternative theory with Robin boundary condition instead of Dirichlet boundary condition if necessary (see [34,38] and the references therein), the proof can be similarly given through those in [[26], pp. 6163].
Case (ii): Since the priori estimate (36) of the solution of the Equation (32) has been established in Lemma 3.5 (ii), we can use the Fredholm's alternative and obtain the unique solution of the Equation (32). □
Let us now introduce the auxiliary nonlinear problems: for m ≥ 1 and 0 ≤ n < N, we find the function ϕ and scalars c_{ij}, i = n + 1, ..., n + m, j = 1, 2, such that
and for m = 0 and 1 ≤ n ≤ N, we find the solution ϕ of the nonlinear Equation (26).
The following result can be proved using standard arguments as in [26,28].
Proposition 3.2 (i) If m ≥ 1 and 0 ≤ n ≤ N, given a fixed number δ > 0, there exist positive numbers ε_{0 }and C such that for any points , l = n + 1, ..., n + m, in Ω' satisfying the relation (33), there is a unique solution for the Equation (55) which satisfies
for all ε < ε_{0 }and t ∈ (0, 1]. Moreover, the map p' → ϕ is C^{1 }and
(ii) If m = 0 and 1≤n≤N, there exist positive numbers ε_{0 }and C such that there is a unique solution for the Equation (26) which also satisfies the estimate (56) for all ε < ε_{0 }and t ∈ (0, 1].
Now we only need to find a solution to the Equation (26) with m ≥ 1 and 0 ≤ n ≤ N, and hence to the Equation (55) if is such that
Let us introduce the energy functional of the Equation (3), namely for t = 1,
and for t ∈ (0,1),
Furthermore, we define
where and with the solution ϕ of the Equation (55).
The finite dimensional variational reduction is meaningful in view of the following property.
Proposition 3.3 If satisfying the relation (33) is a critical point of F_{ε,t }with t ∈ (0, 1], then is a critical point of J_{ε,t}, namely a solution of the Equation (3). Besides, on any compact subsets S of (Ω' \ Γ)^{m }\ Δ_{m }the following expansion holds
where
for ε small.
Proof. Step 1: Let us define for t = 1,
and for t ∈ (0,1),
Then , where . Moreover, for k = n + 1, ..., n + m, l = 1, 2, it holds
Since ϕ(p') is a solution of the Equation (55), v = V(p') + ϕ(p') satisfies
By (62), (63), D_{p}F_{ε,t}(p) = 0 implies for t ∈ (0, 1],
From the definition of V, it can be directly checked , where o(1) is in the sense of the L^{∞}norm for ε small. Since . Hence it follows
which is a strictly diagonal dominant system. This implies that c_{ij }= 0, ∀ i = n + 1, ..., n + m, j = 1, 2. By (63), is a critical point of J_{ε,t}, that is, a solution of the Equation (3).
Step 2: Set . Using DI_{ε,t}(V + ϕ)[ϕ] = 0, a Taylor expansion and an integration by parts, it follows that for t ∈ (0, 1),
and similarly, for t = 1,
Note that ║ϕ║_{∞ }= O(ρlog ε), ║N (ϕ)║_{n,m }= O(ρ^{2} log ε^{2}), ║R║_{n,m }= O(ρ), and ║W║_{n,m }= O(1). Then from (64), (65), it is easy to deduce for t ∈ (0, 1],
Hence, from (66), the expansion (60) satisfies the property (61).□
Finally, we need to write the precisely asymptotical expansion of J_{ε,t}(U). To realize it, we first establish the following result:
Lemma 3.6 Assume that points , satisfy the relation (33), then fort ∈ (0,1) and i = 1, ..., n + m, there hold
and
uniformly in and in for ε small.
where . Since has the gradient estimate on ∂Ω (see [[26], p. 76])
we can easily get . Using the same technique with the proof of Lemma 2.1, we can also get uniformly in and in for ε small, which means (67). From the definition of the regular part of Green function, we can also derive (68). □
Proposition 3.4 The following asymptotical expansions hold for t = 1,
and for t ∈ (0,1),
where φ_{n,m}(p) is defined by (5), and for ε small, Θ_{ε }is a bounded, smooth function of p = , uniformly on points in Ω' satisfying the relation (33).
Proof. According to [[26,28], Lemma 6.1], it only remains to discuss the asymptotical expansion of the energy J_{ε,t}(U) with respect to t ∈ (0,1). By (11), it follows
and
which together with the Equation (13) yields
Furthermore
From (9), (13), and (14), it implies
Then if i ≠ j for any i and j,
and if i = j for any i,
On the other hand, from (21) and (22), it follows
As a result, it derives
Now using the choice for 's by (15), together with (71)(74), it holds
which derives the asymptotical expansion (70) by (5), (67), and (68). □
4 Proofs of theorems
In this section, we carry out the proofs of Theorems 1.1 and 1.2 basing on the finite dimensional reduction. Now we introduce the definition of C^{0}stable critical point of the function φ_{n,m }just like in [28,36,39].
Definition 4.1. We say that p is a C^{0}stable critical point of φ_{n,m }in (Ω' \ Γ)^{m }\ Δ_{m}, which says that if for any sequence of the functions ψ_{j }such that ψ_{j }→ φ_{n,m }uniformly on the compact subsets of (Ω' \ Γ)^{m }\ Δ_{m}, ψ_{j }has a critical point ξ_{j }such that ψ_{j}(ξ_{j}) → + φ_{n,m}.
In particular, if p is a strict local maximum or minimum point of φ_{n,m}, p is a C^{0}stable critical point of φ_{n,m}.
Proof of Theorem 1.1. Case (i): m ≥ 1 and 0 ≤ n ≤ N. Let
where ϕ is the unique solution of the problem (55), which is established in Proposition 3.2. From Proposition 3.3, v(y) is a solution of the Equation (16), namely is a solution of the Equation (3) if satisfying the relation (33) is a critical point of the function F_{ε,t}(p) with t ∈ (0, 1]. This implies that we only need to find a critical point p_{ε }of the following function in (Ω' \ Γ)^{m }\ Δ_{m}
where
From Propositions 3.3 and 3.4, it follows that
where θ_{ε,t}(p) = O(ρ^{2} log ε) for any t ∈ (0, 1], and Θ_{ε}(p) is uniformly bounded on any compact subset S of (Ω' \ Γ)^{m}\ Δ_{m }for ε small. Then , and for any t ∈ (0,1), uniformly on for ε small. By Definition 4.1, there exists a critical point of the function such that , and for any t ∈ (0,1). Moreover, up to a subsequence, there exists such that p_{ε }→ p for ε small, and φ_{n,m}(p*) = φ_{n,m}(p). Hence, is a family of solutions of the Equation (3). As a consequence, from the related properties of U(p_{ε}) and , we easily know that for any λ > 0, u_{ε }is uniformly bounded on , and for ε small.
Finally, we show that u_{ε }satisfies the concentration property:
for ε small. In fact, using the inequality e^{s } 1 ≤ e^{s}s for any and the estimate (56), we obtain
Then from the asymptotical expression (21) and (22) of W(p_{ε})(y), we can easily get
which implies (77).
Case (ii): m = 0 and 1 ≤ n ≤ N. From Proposition 3.2 (ii), we find that is a family of solutions of the Equation (3) with . Then we can get the needed multiple blowingup and concentrating properties of u_{ε }through the similar proof of Case (i). □
In order to give the proof of Theorem 1.2, we need a version of the maximum principle under DirichletRobin boundary conditions, which is the extension of the corresponding one with respect to Dirichlet or Robin boundary condition only.
Lemma 4.2 Assume that T ⊆ ∂Ω is a relatively closed subset, b > 0 is a smooth function on is a smooth function. If u is a solution of the equation
where λ > 0, there exists a constant C(b) > 0 only depending on b(x) such that
Proof. The proof is similar to that of Lemma 2.6 in [33]. □
Proof of Theorem 1.2. Using the maximum principle with DirichletRobin boundary conditions instead of Robin boundary condition if necessary (see Lemma 4.2), the proof can be similarly given out through that of Theorem 1.1. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
This study was supported by N. N. S. F. C. (11171214).
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