Concentrating solutions for an anisotropic elliptic problem with large exponent

We consider the following anisotropic boundary value problem $$\nabla 
(a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, 
\quad u = 0 \ \ \mbox{on} \ \partial\Omega,$$ where $\Omega \subset 
\mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent 
and $a(x)$ is a positive smooth function. We investigate the effect 
of anisotropic coefficient $a(x)$ on the existence of concentrating 
solutions. We show that at a given strict local maximum point of 
$a(x)$, there exist arbitrarily many concentrating solutions.

where Ω is a smooth bounded domain in R 2 , p is a large exponent and a(x) is a smooth positive function over Ω. Problem (1) was motivated by the study of the following equation ∆u + u p = 0, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R 2 , p > 1. Problem (2) has been studied by many people in the last two decades. Standard variational methods have shown the existence of least energy solution. In [23,24] the authors show that the least energy solution u p has bounded L ∞ -norm and u p ∞ is bounded away from zero uniformly in p, for p large. In [1,21] the authors give a further description of the asymptotic behavior of u p , as p → ∞, by identifying a limit profile problem of Liouville-type: ∆u + e u = 0 in R 2 , and showing that u p ∞ → √ e as p → ∞. For singular limits in Liouville-type equation, readers may refer to [10]. Conversely, many people are interested in Also the authors in [12] compared problem (2) with some widely studied problems which have some analogies with it. For example, ∆u + u N +2 N −2 −ε = 0, u > 0 in Ω, u = 0 on ∂Ω, which has been studied by [3,15,17,25,26]. For more details readers may refer to [12] and references therein. On the other hand, the case of sign changing solutions to problem (2) has been considered in [13].
Our motivation in problem (1) are twofolds. First, since problem (1) is a natural generalization of equation (2), one may expect similar results in [12] hold. In fact, this is true for general domain Ω whether it is simply connected or not. Secondly, problem (1) is a special case of problem (2) in higher-dimension N ≥ 3. Actually, when we work with the cross-section of an N -dimensional torus having axial symmetry, we can find that problem (2) is reduced to (1): let the torus be T = (x , x N ) : ( x − 1) 2 + x 2 N ≤ r 2 0 with x = (x 1 , . . . , x N −1 ), x = x 2 1 + · · · + x 2 N −1 , r 0 < 1.
If we look for solutions in the form u(x , x N ) = u(r, x N ) with r = x for (2), a direct calculus shows that the problem is transformed to ∇(r N −2 ∇u) + r N −2 u p = 0 in Ω = (r, x N ) : (r − 1) 2 + x 2 N < r 2 0 with u = 0 on ∂Ω. This is just the problem (1) with a(r, x N ) = r N −2 .
When we consider the generalized problem (1), there are some natural questions: Q1. Can we move the topological conditions on Ω for non-constant function a(x)? Q2. Is there any solution with concentrating points not simple?
In [14], for Hénon equation ∆u + |x| 2α u p = 0 in B(0, 1), u = 0 on ∂B(0, 1), where α ∈ N, B(0, 1) is a unit ball in R 2 with radius 1 and center 0, the authors find many positive solutions and sign changing solutions for p large enough. Due to the function |x| 2α , it is not necessary that the domain is not simply connected. But the solutions they construct concentrate at simple symmetric points, hence there is no clue to Q2.
In this paper we answer these two questions affirmatively. Our main result is the following: CONCENTRATING FOR AN ANISOTROPIC PROBLEM 3 Theorem 1.1. Let x 0 ∈ Ω be a strict local maximum point of a(x), i.e. there exists a neighborhood B(x 0 , δ), δ > 0 such that Then for any m ∈ N * , problem (1) has a family of solutions u p such that as p → +∞, such that for any ρ > 0, u p → 0 uniformly in Ω\ ∪ m j=1 B(ξ p j , ρ) and Remark 1. In Theorem 1.1, if we have the following expansion of a at x 0 : in a neighborhood of x 0 , then the distance between concentrating points satisfies This implies that the flatter the anisotropic coefficient is, the bigger is the distance between the bubbles.
Since we will cite the results in [27] in the following proof, it is necessary to introduce the work in [27] quickly. Namely, in [27], Wei, Ye and Zhou have studied the anisotropic Emden-Fowler equation where a(x) is a smooth positive function in Ω. It is easy to see that problem (5) is a natural generalization of the following classical Emden-Fowler equation, or Gelfand's equation which has been studied very widely, see [2,16,18,20,22,28] and the references therein. They proved that if a(x) has a local strict maximum point x 0 , then for any m ∈ N * problem (5) has a family of solutions u ε which makes an m-bubbles concentration at x 0 . Theorem 1.1 is proved via the so-called " localized energy method "-a combination of Liapunov-Schmidt reduction method and variational techniques. Namely, we first use Liapunov-Schmidt reduction method to reduce the problem to a finite dimensional one, with some reduced energy. Then, the solutions in Theorem 1.1 turn out to be generated by critical points of the reduced energy functional. Such an idea has been used in many other papers. See for instance [4,8,9,11,12,27] and the references therein. Here we will follow those of [12] and [27].
Throughout the paper, the symbol C always denotes a positive constant independent of p, which could be changed from one line to another and | · | is for Euclidean norm in R 2 .

2.
Ansatz for the solutions. The purpose of this section is to provide an ansatz for problem (1) and give some basic estimates for the error term.
It is well known that the solutions to the following Liouville-type equation (see [7]) can be all written in the following form Let and G(x, y) be the Green's function satisfying ∆ a G(x, y) + 8πδ y = 0 in Ω, G(x, y) = 0 on ∂Ω.
Given now ξ j ∈ Ω, with (7) we define where δ j = µ j e − p 4 and µ j is to be determined later. The configuration space for (ξ 1 , · · · , ξ m ) is chosen as follows where M is given by Note that by the choice of ξ j , we have if (ξ 1 , · · · , ξ m ) ∈ Λ, However, U δj ,ξj is a good first approximation, but not enough for our approximation. We need to refine this first approximation.
Let us call v ∞ (y) = U 1,0 (y) and radial functions w 0 , w 1 solving According to [5], for a radial function f (y) = f (|y|) there exists a radial solution We state the following lemma whose proof is given in [12] on page 37-38.
where C 0 = 12 − 12 log 2. More precisely, we have the exact expression of w 0 , and for a suitable constant C 1 .
Define now for any x ∈ Ω, where Then Lemma 2.3. For any β ∈ (0, 1), p large enough, H p j (x) = 1 uniformly in Ω, where H is the regular part of Green's function defined by (9).
Proof. The boundary condition satisfied by H p j (x) is The regular part of Green's function H(x, y) satisfies For the difference, let Z p (x) := in Ω and According to the definition of U j in (18), we get by direct computation that , applying polar coordinates with center ξ j , i.e. r = |x − ξ j |, there holds In conclusion, for any 1 < q < 2, we have Applying L q theory, By Sobolev embedding theorem, we obtain Observe that |y| ≤ 1 and for any i = j, is a good approximation for a solution to problem (1) provided that Lemma 2.4. Let where C > M is large enough but fixed independent of p, then system (24) is solvable in Σ.

LIPING WANG AND DONG YE
Proof. Let us consider the following vector function Obviously g(0; µ) = 0 is solvable for all C > 0, that is, T = ∅. It's easy to see that T is closed. Now we would prove that T is open. If so, T = [0, 1] which tells us that (24) is solvable. Indeed, for any µ ∈ Σ, then Using the expansion of exponential function, we can get that Direct computation gives out that and for any i = j, So ∇ µ g(t; µ) is not singular over Σ for any t ∈ [0, 1] and large p.
For any t 0 ∈ T with g(t 0 ; µ 0 ) = 0 and µ 0 ∈ Σ, using Implicit Function Theorem, we see that g(t; µ) = 0 is solvable in some open neighborhood of (t 0 , µ 0 )), that is, A direct computation shows that, for p large, µ = (µ 1 , . . . , µ m ) satisfies Observe that µ j may not be O(1) since ξ i → x 0 for all i = 1, . . . , m, but we can derive that Remark 2. For p large, from the above computations, we can get easily that for with p large enough. By (23) we obtain that 0 < U ξ ≤ 2 √ e and for any ρ > 0, using the property of δ j and µ j , we find that By maximum principle, G(x, ξ j ) > 0 in Ω. In conclusion, Let us set and we introduce the following functional defined in H 1 0 (Ω): whose nontrivial critical points are solutions to (1). It is easy to see that problem (1) is equivalent to by maximum principle. We will look for solutions u of problem (1) in the form u = U ξ + φ, where φ will represent a higher-order term in the expansion of u.
Observe that and In terms of φ, problem (28) becomes The main step in solving problem (31) is that of a solvability theory for the linear operator L under a suitable choice of the points ξ i . In developing this theory, we will take into account the invariance, under translations and dilations, of the problem ∆v + e v = 0 in R 2 . We will perform the solvability theory for the linear operator L in weighted L ∞ -norm space. For any h ∈ L ∞ (Ω), define We conclude this section by showing an estimate of R ξ in · * .
Proposition 1. There exist C > 0 and p 0 > 0 such that for any ξ ∈ Λ and p ≥ p 0 we have Proof.

Thus in this region
Finally, for any i = 1, . . . , m, |x − ξ i | ≥ p −2M , then as the computations in (34) and Remark 2, which leads to the end of proof.
It is well known that any bounded solution to is a linear combination of z i , i = 0, 1, 2. See Lemma 2.1 of [6]. Now we consider the following linear problem: given h ∈ C(Ω), find a function φ ∈ H 2 (Ω) such that for some coefficients c ij , (i = 1, 2, 1 ≤ j ≤ m). Here and in the sequel, for any i = 0, 1, 2 and j = 1, . . . , m, we denote The main result of this section is the following: Proposition 2. There exist p 0 > 0 and C > 0 such that for any p > p 0 , ξ ∈ Λ, h ∈ C(Ω) there is a unique solution to problem (41), which satisfies Proof. The proof consists of six steps.
In order to prove this fact, we show the existence of a positive function Z in Ω satisfying L[Z] < 0. We define Z to be Clearly Φ 0 ≥ 2 in Ω and bounded. Thus in Ω, 1 ≤ Z(x) ≤ C where C is independent of R. On the other hand in Ω, Then for p large, is what we are looking for.
Step 2. Let k be defined in Step 1. Let us define the " inner norm " of φ in the following way We claim that there is a constant for any h ∈ C 0,α (Ω). We will establish this estimate with the use of barriers. Indeed, where Z(x) was defined in the previous step, then on ∂ Ω, φ ≥ |φ| and the above computation shows that L[ φ] ≤ −|h|. By maximum principle we get By the definition of φ i , we obtain for some constant C independent of h.
By Lemma 3.1 and a(x) ∈ C 1 (Ω), elliptic estimates readily imply that φ n j (y) converges uniformly over compact sets to a bounded solution φ ∞ j of This implies that φ ∞ j is a linear combination of the functions z i , i = 0, 1, 2. Since φ n j (y) ∞ ≤ 1, by Lebesgue's theorem, the orthogonality conditions on φ n j pass to the limit and give Hence φ ∞ j ≡ 0 for any j = 1, . . . , m contradicting to lim inf n→∞ φ n i > 0.
Step 4. We prove that there exists a positive constant C such that any solution φ to L[φ] = h in Ω, φ = 0 on ∂Ω and in addition the orthogonality conditions: Ω e U δ j ,ξ j Z ij φ = 0 for i = 1, 2, j = 1, . . . , m, Proceeding by contradiction as in Step 3, we can suppose further that p n h n * → 0 as n → ∞.
But here we loss the limit condition Hence we have that for some constant C j . To reach a contradiction, we have to show that C j = 0 for any j = 1, . . . , m. We will obtain it from the stronger condition (45) on h n . To this end, we perform the following construction. By Lemma 2.2, there exist radial solutions w and ζ respectively of equations For simplicity, frow now on we will omit the dependence on n. For j = 1, . . . , m, let Notice that |v j (x)| ≤ C log( |x−ξj | δj + 2) + C| log δ j |. Suppose h j satisfy the equation Thus h j (x) = − 1 3 H(x, ξ j ) + O(δ j ), whose proof is very similar to Lemma 2.3, so we omit it. Let Also we can find that where
Necessarily, C j = 0 by contradiction and the claim is proved.
Step 5. We establish the validity of the following estimate: for the solutions of problem (41) and h ∈ C 0,α (Ω).
Step 4 gives |c ij | since e U δ j ,ξ j Z ij * ≤ 2 e U δ j ,ξ j * ≤ 16. Arguing by contradiction of (56), we can proceed as Step 3 and suppose further that We omit the dependence on n. It suffices to estimate the values of constants c ij . For i = 1, 2 and j = 1, . . . , m, now we define Γ ij as the following According to [12] on page 51, we have and |T ij | ≤ |Z ij | + Cδ j ≤ 2 + Cδ j ≤ 3.
Multiply the first equation of (41) by a(x)Γ ij and integrate by part, we get Since ∆Γ ij = ∆Z ij = −e U δ j ,ξ j Z ij , the above equality can be changed to First, let us deduce the following " orthogonality " relations: for 1 ≤ i, l ≤ 2 and , where δ il denotes the Kronecker's symbol. where Inserting the estimates (60) and (61) into (58), we deduce that Hence we obtain that Obviously we get 2 l=1 m h=1 |c lh | = o(1).

As in
Step 4, there holds for some j and constant C j . Hence, in (61) we have a better estimate Therefore, we get that the R.H.S. of (58) = o p −1 , and in turn, By Fredholm's alternative theorem, it is equivalent to the uniqueness of solutions to this problem, which is guaranteed by Proposition 2. Moreover, by elliptic regularity theory this solution is in H 2 (Ω). As p > p 0 fixed, by density of C 0,α (Ω) in (C(Ω), · ∞ ), we can approximates h ∈ C(Ω) by smooth functions and, by elliptic regularity theory, we can show that (42) holds for any h ∈ C(Ω). This ends the proof.
Using the theory developed in the previous section for the linear operator L, we have Lemma 4.1. There exist C > 0 and p 0 > 0 such that, for any p > p 0 and ξ ∈ Λ, problem (63) has a unique solution φ ξ satisfying Furthermore, the function ξ → φ ξ is C 1 .
Proof. The proof of this lemma can be done along the lines of that of Lemma 4.1 in [12]. We omit the details.
Proof. The proof is very similar to that of Lemma 5.1 in [12]. We omit it here.
Next lemma shows the leading term of the function F p (ξ).