Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions

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 , Ω is a bounded, smooth domain in , Γ := {p₁, ..., 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.

In this article, we mainly investigate the mixed boundary value problem:

for ε small, where , Ω is a bounded, smooth domain in , Γ:= {p₁, ..., 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 [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 G₁ = G_1,ε and H₁ = H_1,ε. Since ε exactly disappears in the Equation (2)|_{t = 1}, G₁ and H₁ 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 = (p₁, ..., 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 [7,17-23]). 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 of the function φ_n,m with n = 0 generates a family of the solutions u_ε which blow-up 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 non-degenerate 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 blowing-up 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 C⁰-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, α₀), where α₀ = min{0, α₁, ..., 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 blowing-up 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 C⁰-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 n-different points in Γ, and m-points 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 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 . 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 n-different points in Γ, and m-points 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 Liouville-type 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 blowing-up 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.

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 {k₁, ..., 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

where (x) is the solution of

Then satisfies

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 multi-index 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,

and if for some i,

On the other hand, if for all i = 1, ..., n + m, obviously,

and if for some i,

Now from (14), (15) and (17), we have

In summary, we set

and if for all i = 1, ..., n + m,

while for some i,

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

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_tϕ = 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

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 , the function

with any and , is the solution of -∆u = |z|^2αe^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)ϕ₀(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, non-increasing cut-off function such that for a large but fixed number R₀ > 0, χ(r) = 1 if r ≤ R₀, and χ(r) = 0 if r ≥ R₀ + 1. Additionally, set α₀ = min{0, α₁, ..., α_N}. For any α ∈ (-1, α₀), 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 ε₀ 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 < ε₀ and t ∈ (0, 1]. Moreover, the map p' ↦ ϕ is C¹ and

where .

(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε₀ and C such that there is a unique solution ϕ ∈ L^∞(Ω_ε) of the Equation (32), which satisfies

for all ε < ε₀ 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 ε₀ 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 < ε₀.

(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε₀ 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 < ε₀.

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_iϕ = 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)²a^-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 R₁ > 0 such that with . Consider and let us translate and rotate Ω_ε so that q' = 0 and Ω_ε approaches the upper half-plan . 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 ε₀ 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 < ε₀.

(ii) If m = 0 and 1 ≤ n ≤ N, there exist positive numbers ε₀ and C such that for any solution ϕ of the Equation (32) with t ∈ (0, 1], one has

for all ε < ε₀.

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 > R₀ + 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 η₁(r) and η₂(r) are smooth cut-off functions with the properties: η₁(r) = 1 for r < R, η₁(r) = 0 for ; η₂(r) = 1 for r < δ, η₂(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. 61-63].

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 ε₀ 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 < ε₀ and t ∈ (0, 1]. Moreover, the map p' → ϕ is C¹ and

where and .

(ii) If m = 0 and 1≤n≤N, there exist positive numbers ε₀ and C such that there is a unique solution for the Equation (26) which also satisfies the estimate (56) for all ε < ε₀ 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_pF_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(ρ|log ε|), ║N (ϕ)║_n,m = O(ρ²| 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.

Proof. Set . Then satisfies

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). □

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 C⁰-stable critical point of the function φ_n,m just like in [28,36,39].

Definition 4.1. We say that p is a C⁰-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 C⁰-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(ρ²| 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 blowing-up 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 Dirichlet-Robin 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 Dirichlet-Robin 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. □

The authors declare that they have no competing interests.

All authors typed, read and approved the final manuscript.

This study was supported by N. N. S. F. C. (11171214).

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