Abstract
In this article, we mainly construct multiple blowing-up and concentrating solutions for a class of Liouville-type equations under mixed boundary conditions:
for ε small, where
2000 Mathematics Subject Classification: 35B25; 35J25; 35B38.
Keywords:
multiple blowing-up and concentrating solution; Liouville-type 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
Such problems occur in conformal geometry [1], statistical mechanics [2-4], Chern-Simons vortex theory [5-11] and several other fields of applied mathematics [12-16]. In all these contexts, an interesting point is how to construct solutions which exactly "blow-up" 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 blowing-up 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
Thus, we consider the more general model problem:
where
It is known that for
in the sense of measures in
(see [7,17-23]). An obvious problem for the Equation (3) is the reciprocal, namely the existence of multiple blowing-up solutions with concentration points near critical points of φ_{n,m}.
The earlier result concerning the existence of multiple blowing-up 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 non-degenerate critical point
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 blowing-up and concentrating solutions for the Equation
(3)|_{t = 1 }with
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
Theorem 1.1 Let 0 ≤ n ≤ N and
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 blowing-up and concentrating solutions. Based on this point, we also consider the Dirichlet-Robin boundary value problem:
where T ⊆ ∂Ω is a relatively closed subset and
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 n-different points
Finally, it is very interesting to mention that to prove the above results we need to choose the classification solutions of the following Liouville-type equation to construct concentrating solutions of the Equation (1) or (3):
given by
with
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
where
Then
Lemma 2.1 For t ∈ (0, 1] and
uniformly in
Proof. Set
where
For any t ∈ (0, 1], it is easy to check
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 multi-index α with |α| ≤ 2, which derives (14) uniformly in
From this lemma we can construct the approximate solution
Consider the scaling of the solution of the Equation (3)
then v(y) satisfies
where
and
Obviously,
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
and if
On the other hand, if
and if
Now from (14), (15) and (17), we have
In summary, we set
and if
while
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 well-known 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
Remark 3.1 These properties of the operator L_{i }have been discussed in the above papers only if
Lemma 3.2 For
is proportional to
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
with any
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,
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, non-increasing cut-off 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
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
there is a unique solution
for all ε < ε_{0 }and t ∈ (0, 1]. Moreover, the map p' ↦ ϕ is C^{1 }and
where
(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
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
Proof. Case (i): First consider the "inner norm"
We will establish it with the help of suitable barrier.
Consider that the function
we define a bounded comparison function
with a > 0. Set
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,
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
We prove the priori estimate (38) by contradiction. Assume that there exist a sequence
ε_{k }→ 0, points
Then from the estimate (41), ║ϕ_{k}║_{l }≥ κ or ║ϕ_{k}║_{o }≥ κ for some κ > 0. Briefly set ε:= ε_{k},
for z ∈ B_{R}(0). Obviously, for any
From Lemma 3.2, this equation implies that
If ║ϕ_{k}║_{o }≥ κ and ║ϕ_{k}║_{l }→ 0, there exists a point q ∈ ∂Ω and a number R_{1 }> 0 such that