Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data

Let Ω be a bounded domain in \begin{document}$\mathbb{R}^2 $\end{document} with smooth boundary, we study the following Neumann boundary value problem \begin{document}$\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$ \end{document} where \begin{document} $ν$ \end{document} denotes the outer unit normal vector to \begin{document} $\partial \Omega$ \end{document} , \begin{document} $h∈ C^{0,α}(\partial \Omega)$ \end{document} , \begin{document} $s>0$ \end{document} is a large parameter and \begin{document} $\phi_1$ \end{document} is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of \begin{document} $\phi_1$ \end{document} on the boundary as \begin{document} $s\to+∞$ \end{document} .

where ν denotes the outer unit normal vector to ∂Ω, h ∈ C 0,α (∂Ω) is given, s > 0 is a large parameter and φ 1 is a positive first eigenfunction of the Steklov problem (see [2]): (1.2) We denote eigenvalues of (1.2) as 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · , and set ρ(x) ∈ H 1 (Ω) as a unique solution of    − ∆ρ + ρ = 0 in Ω, where ε > 0 is a small parameter. Elliptic equations with this type of nonlinear Neumann boundary condition arise in conformal geometry (prescribing Gaussian curvature of the domain and curvature the boundary), see [21], and also in corrosion modelling which arises from electrochemistry, see [18,25]. For problem (1.5), Dávila, del Pino and Musso in [12] have analyzed the asymptotic behavior of families of solutions u ε with positive, uniformly bounded mass ε ∂Ω e uε as ε → 0. It turns out that, up to subsequences, there is an integer m ≥ 1, such that and its regular part H(x, y) = G(x, y) + 2 log |x − y|. (1.8) Reciprocally, the authors in [12] also proved the existence of families of boundary bubbling solutions u ε to (1.5) with above properties. More precisely, they showed that, given any integer m ≥ 1, problem (1.5) has at least two distinct families of solutions u ε for which (1.6) holds, and the peaks of these two solutions are located around two distinct critical points of a certain functional of m points of the boundary.
Here we introduce a natural generalization of equation (1.5), namely the anisotropic elliptic problem    − ∇(a(x)∇u) + a(x)u = 0 in Ω, where a(x) is a smooth positive function over Ω. In [32], the author investigated the asymptotic behavior of solutions u ε to (1.9) for which ε ∂Ω e uε is bounded, and BOUNDARY BUBBLING SOLUTIONS 5469 showed that, up to subsequences, it will develop a finite number of bubbles ξ i ∈ ∂Ω such that for some integers m i , k ≥ 1, and these bubbling points are nothing but critical points of the function a(x) on the boundary. Thus in contrast with the above work in [12], a natural problem is posed that whether (1.9) does admit a boundary bubbling solution such that m i > 1. Let us point out that m i = 1 corresponds to a simple blow up at ξ i while m i > 1 gives rise to a non-simple (or multiple) blow up. Indeed, the authors in [31] gave a positive answer to this reciprocal question through the construction of solutions to (1.9) with a cluster of multiple bubbles around any isolated local maximum point of a(x) on the boundary. This paper is devoted to studying the existence and concentration behavior of boundary bubbling solutions to problem (1.4) (or (1.1)). We prove that there exists a family of solutions of problem (1.4) with the accumulation of arbitrarily many boundary bubbles around maximum points of φ 1 on the boundary. Our main result can be stated as follows.
Theorem 1.1. Let Λ be a subset of ∂Ω satisfying sup ∂Λ φ 1 < sup Λ φ 1 . Then given any positive integer m, there exists t m > 0 such that for any t > t m , there is a solution u t of problem (1.4) which satisfies lim t→+∞ ∂Ω k(x)e −tφ1 e ut = 2mπ.
From Theorems 1.1 and 1.2 we see that problem (1.4) has a solution that exhibits the phenomenon of multiple blow up at boundary maximum points of φ 1 , in particular we have the concentration properties with some integers m i > 1, where ξ i 's are boundary maxima of φ 1 . Let us remark the interesting analogy between these results and those known for the following elliptic problem of Ambrosetti-Prodi type [1]: where Ω is a bounded smooth domain in R 2 , h ∈ C 0,α (Ω), s > 0 is a large parameter and φ 1 is a positive first eigenfunction of the problem −∆φ = λφ under Dirichlet boundary condition in Ω. In the early 1980s Lazer and McKenna conjectured that equation (1.10) has an unbounded number of solutions as s → +∞ (see [19]). If we denote its eigenvalues as 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · , and set ρ( where k(x) = e −ρ(x) and t = s/λ 1 . In [15], del Pino and Muñoz gave a positive answer to the Lazer-McKenna conjecture by constructing solutions of problem (1.11) with the following asymptotic profile where m i > 1, ξ i 's are inner maxima of φ 1 and G D (x, ξ i ) denotes the Green function of the problem Surprisingly enough, this multiple bubbling phenomenon of solutions of problem (1.11) contrasts with that of the Liouville equation where Ω is a bounded smooth domain in R 2 , ε > 0 is a small parameter and k(x) is a nonnegative, not identically zero function of class C 2 (Ω). Indeed, the results in [5,20,24,29] shows that the bubbling of solutions of equation (1.12) with positive, uniformly bounded mass ε 2 Ω k(x)e u should be simple, namely all m i 's are equal to one. Reciprocally, solutions of (1.12) with this bubbling behavior have been obtained in [3,6,14,17]. In particular, the construction of solutions with an arbitrarily large number of bubbling points was achieved in [14] in the case that Ω is not simply connected. Finally, it is necessary to point out that multiple bubbling has also been built in [35] for solutions of the two-dimensional anisotropic Emden-Fowler equation around any isolated local maximum point of the uniformly positive, smooth function a(x). We mention that the Lazer-McKenna conjecture holds true for some Ambrosetti-Prodi problems with other type of nonlinearities where Ω is a bounded smooth domain in R N , N ≥ 3, s > 0 is a large parameter, φ 1 > 0 is an eigenfunction of −∆ with Dirichlet boundary condition corresponding to the first eigenvalue λ 1 , and g : R → R is a continuous function such that lim t→−∞ g(t) t = α < lim t→+∞ g(t) t = β and (α, β) contains some eigenvalues of −∆ subject to Dirichlet boundary condition. Here α = −∞ and β = +∞ are allowed. The reader can refer to [26,27,28] for the asymptotically linear case g(t) = βt + − αt − involving sufficiently large β, to [9,10] for the subcritical case g(t) = t p + +λt, λ < λ 1 , [16,22,23,34] for the critical case g(t) = t N +2 N −2 + + λt, 0 < λ < λ 1 and N ≥ 6, to [7] for the superlinear nonhomogeneous case g(t) = t p + + t q − , 1 < q < p < N +2 N −2 and N ≥ 4, to [4,11,13] for the superlinear homogeneous case g(t) = |t| p , 1 < p < N +2 N −2 if N ≥ 3, 1 < p < +∞ if N = 2, and to [8,33] for other cases, where t + = max{t, 0} and t − = max{−t, 0}.
The proof of our results relies on a very well-known Lyapunov-Schmidt reduction procedure. In Section 2 we exactly describe the ansatz for the solution of problem (1.4) and rewrite problem (1.4) in terms of a linearized operator for which a solvability theory, subject to suitable orthogonality conditions, is performed through solving a linearized problem in Section 3. In Section 4 we solve an auxiliary nonlinear problem. In Section 5 we reduce the problem of finding boundary bubbling solutions of (1.4) to that of finding a critical point of a finite-dimensional function. In the last section we write an asymptotic expansion for the energy functional appeared in Section 5, and further give the proof of Theorem 1.1.
Throughout this paper, unless otherwise stated, the letters c, C will always denote various universal positive constants that are independent of t and m for t sufficiently large.
2. Preliminaries and ansatz for the solution. In this section we will provide an ansatz for solutions of problem (1.4). It is well known (see [21,30,36]) that any solution of the problem where R 2 + denotes the upper half-plane {(x 1 , x 2 ) : x 2 > 0} and ν the unit exterior normal to ∂R 2 + , must be of the form where τ ∈ R and µ > 0 are parameters. Set (2. 3) The basic cells for the construction of an approximate solution of equation (1.4) are the one-parameter family of functions ψ µ . An important fact that we will use later on is the nondegeneracy of ψ µ up to the natural invariance of equation (2.1) under translations and dilations, ζ 1 → ψ µ (x 1 − ζ 1 , x 2 ) and κ → ψ µ (κx) + log κ. Thus we set and which exactly correspond to variations of ψ µ,τ defined in (2.2) along its parameters of dilation and translation, respectively. Obviously, these objects lie in the kernel of the linearization of equation (2.1) at the solution ψ µ , namely they solve the problem (2.6) Reciprocally, any bounded solution of (2.6) is a linear combination of z µ0 and z µ1 , see [12] for a proof. Let us fix a subset Λ of ∂Ω as in the statement of Theorems 1.1-1.2. For the sake of convenience we assume sup x∈Λ φ 1 (x) = 1. (2.7) The configuration space for m-tuple ξ = (ξ 1 , . . . , ξ m ) we choose is the following where β is given by Observe that the function = εe u on ∂R 2 + . Thus, for numbers µ i > 0, i = 1, . . . , m, yet to be chosen, we define where We hope to take m i=1 u i (x) as an initial approximate solution of (1.4). So we modify it to be where (2.13) (2.14) Lemma 2.1. For any 0 < α < 1, ξ = (ξ 1 , . . . , ξ m ) ∈ O t , then we have uniformly in Ω, where H is the regular part of Green's function defined in (1.8).
Proof. First, on the boundary, we have On the other hand, the regular part of Green's function H(x, ξ i ) satisfies Following Lemma 3.1 from [12], we can easily prove that for 1 < p < 2, and for p > 1,
We now hope that for each i = 1, . . . , m, the remainder U − u i = H i + j =i (u j + H j ) vanishes at the main order near ξ i , which can be realized by choosing the parameter µ i such that (2.17) We thus fix µ i a priori as a function of ξ in O t and write µ i = µ i (ξ) for all i = 1, . . . , m. Since ξ = (ξ 1 , . . . , ξ m ) ∈ O t , there exists a constant C > 0 independent of t such that for any t large enough, Consider the scaling of the solution to equation (1.4) We write ξ i = ξ i /ε and define the initial approximate solution to (2.21) as where U is defined by (2.12). Moreover, set 24) and the "error term" Let us see how well ∂V (y) ∂ν matches with W (y) through V (y) so that the "error term" R(y) is sufficiently small for any y ∈ ∂Ω t . Assume first |y − ξ i | ≤ δ/(2εt β ) for some index i and a small constant δ > 0. Then we have and by (2.10) and (2.22), From (2.15) and the fact that H is C 1 (∂Ω 2 ) we get that for any y ∈ Ω t , Hence for |y − ξ i | ≤ δ/(2εt β ),

HAITAO YANG AND YIBIN ZHANG
where the last equality is due to the choice of µ i in (2.17). Therefore when |y − ξ i | ≤ δ/(2εt β ), which, together with (2.25)-(2.26), concludes that in this region On the other hand, if |y − ξ i | > δ/(2εt β ) for all i, by (2.10) and (2.15) we obtain In the rest of this paper, we try to find a solution of problem (2.21) in the form ω = V + φ, where φ will represent a lower order correction. In terms of φ, problem where the "nonlinear term" N (φ) is given by 3. Solvability of a linear problem. In this section we shall study the solvability of the following linear problem: given h ∈ L ∞ (∂Ω t ) and points ξ = (ξ 1 , . . . , ξ m ) ∈ O t , we find a function φ, and scalars c 1 , . . . , c m , such that where W = q(y, t)e V satisfies (2.30) and (2.33), and Z 1i , χ i are defined as follows: let Z 0 , Z 1 denote the functions z µ0 and z µ1 , respectively given by (2.4) and (2.5) with the parameter µ = 1. More precisely, . Furthermore, let d > 0 be a small but fixed radius, depending only on the geometry of Ω, such that M is an open neighborhood of the origin. We can select H i so that it preserves area. Then for any i = 1, . . . , m, j = 0, 1, we define Besides, we consider R 0 a large but fixed positive number and χ(r) a radial smooth, non-increasing cut-off function with 0 ≤ χ(r) ≤ 1, χ(r) = 1 for r ≤ R 0 and χ(r) = 0 for r ≥ R 0 + 1. Let Equation (3.1) will be solved for h ∈ L ∞ (∂Ω t ), but we need to estimate the size of the solution by introducing the following norm: where 0 < σ < 1 is fixed and chosen later on.
Proposition 3.1. Let m be a positive integer. Then there exist constants t m > 1 and C > 0 such that for any t > t m , any points ξ = (ξ 1 , . . . , ξ m ) ∈ O t and any The proof of this result will be divided into four steps which we state and prove next.
Step 1. Constructing a suitable barrier.
Lemma 3.1. There exist positive constants R 1 and C, independent of t, such that for any t > 1 sufficiently large, and 0 < σ < 1, there is HAITAO YANG AND YIBIN ZHANG smooth and positive so that Moreover, we have a uniform bound Observing that ψ 0 is uniformly bounded in Ω t , it is directly checked that, choosing the positive constant C 1 larger if necessary, ψ verifies the required conditions.
Step 2. Handing a linear equation. We consider first the linear equation where we use · * ,∂Ωt to estimate h ∈ L ∞ (∂Ω t ), and for f ∈ L ∞ (Ω t ) we introduce the following norm: For the solution of (3.9) satisfying orthogonality conditions with respect to Z 0i and Z 1i , we have the following a priori estimate.
Lemma 3.2. There exist R 0 > 0 and t m > 1 such that for any t > t m and any solution φ of (3.9) with the orthogonality conditions Proof. Take R 0 = 2R 1 , R 1 being the constant of Lemma 3.1. Thanks to the barrier ψ of that lemma, we deduce the following maximum principle: . Let f , h be bounded and φ a solution to (3.9) satisfying (3.11). Consider the "inner norm" for some constant C independent of φ and t. We prove the lemma by contradiction. Assume that there are sequences of parameters t n → +∞, points ξ n = (ξ n 1 , . . . , ξ n m ) ∈ O tn , functions f n and h n , and corresponding solutions φ n of equation (3.9) with the orthogonality conditions (3.11) such that φ n L ∞ (Ωt n ) = 1, h n * ,∂Ωt n → 0, f n * * ,Ωt n → 0 as n → +∞. (3.14) Using the expansion of W n in (2.30) and elliptic regularity, we can deduce thatφ n i : . . , m}, converges uniformly over compact sets to a bounded solution φ ∞ i of equation (2.6)| µ=1 , which satisfies However, by the result of [12], any bounded solution of equation (2.6)| µ=1 can be expressed as a linear combination of Z 0 and Z 1 . Therefore, (3.15) impliesφ ∞ i = 0 or lim n→+∞ φ n i = 0. But (3.13) and (3.14) tell us lim inf n→+∞ φ n i > 0, which is a contradiction.
Step 3. Establishing an a priori estimate for solutions to (3.9) with the orthogonality condition Ωt χ i Z 1i φ = 0 only.
From estimate (2.18) and definitions (2.11), (2.20) and (2.28) we find and Let η 1 and η 2 be radial smooth cut-off functions in R 2 such that and Given φ satisfying (3.9) and (3.16), we define Remark that the map H ε i preserves area and Z 0i coincides with Z 0i in the region {y ∈ Ω t : |H ε i (y)| ≤ γ i R}, which imply that Z 0i is orthogonal to χ i Z 1i for each i = 1, . . . , m. Furthermore, using definition (3.24), orthogonality condition (3.25) for j = 1 and the fact that Notice that by (3.21) and (3.23), and Ωt We only need to consider d i . Testing definition (3.24) against χ k Z 0k , we obtain a system of (d 1 , . . . , d m ), We denote H the coefficient matrix of system (3.28). By the above estimates, it is clear that P −1 HP is diagonally dominant and thus invertible, where P = diag(γ 1 , . . . , γ m ). Hence H is also invertible and (d 1 , . . . , d m ) is well defined. Estimate (3.17) is a direct consequence of the following three claims.

HAITAO YANG AND YIBIN ZHANG
Furthermore, using the definition of φ again and the fact that the Lemma 3.3 then follows from Claims 1-3 and estimate (3.33).

HAITAO YANG AND YIBIN ZHANG
Proof of Claim 3. Testing (3.32) against Z 0i and using estimates (3.33)-(3.34), we find where we have applied the following two inequalities: From estimates (3.27), (3.29) and (3.30), it follows that for any i = 1, . . . , m, (3.57) To achieve the estimates of d k and e k in (3.31), we have the following claim.
Claim 4. If δ is sufficiently small, but R is sufficiently large, Indeed, once Claim 4 is proven, then substituting (3.58) and (3.59) into (3.57) we conclude and then, by (2.11), Finally, using estimate (3.27) we derive that Proof of Claim 4. Let us first establish the validity of the a priori estimate (3.58). We decompose From (3.42), we get From (3.47) and (3.49), we have (3.64) As for I 2 , we can easily get

and this implies
Ωt On the other hand, by (3.36) we also compute To estimate B i , we decompose supp(η 2i ) ∩ ∂Ω t to some pieces: . . , m, Observe that, by (2.8) and (3.37), and on B 1k , k = i, by (3.79), But on B 2 , thanks to (2.33), Besides, a direct computation shows that for t sufficiently large, and then This combined with (3.76) implies Let us consider the Hilbert space By Fredholm's alternative this is equivalent to the uniqueness of solutions to this problem, which in turn follows from estimate (3.75).
To prove solvability of problem (3.1), let Y i ∈ L ∞ (Ω t ), c i1 , . . . , c im ∈ R be the solution of equation (3.74) with h = χ i Z 1i , namely, (3.83) From the above argument there exist a unique solution Y i ∈ L ∞ (Ω t ), and c ik ∈ R, k = 1, . . . , m, to this equation such that Let us first claim that there exist a constant A, independent of t, such that which concludes the validity of expansion (3.85). Hence the matrix D with entries γ k c ik is invertible for sufficiently large t and D −1 ≤ C uniformly on t. Now, given h ∈ L ∞ (∂Ω t ) and points ξ = (ξ 1 , . . . , ξ m ) ∈ O t , we have a function φ 1 ∈ L ∞ (Ω t ) and scalars f 1 , . . . , f m ∈ R as the solution of equation (3.74), and further define Remark 3.1. A slight modification of the above proof also shows that for any h ∈ L ∞ (∂Ω t ) and f ∈ L ∞ (Ω t ) the equation has a unique solution φ, c 1 , . . . , c m . Moreover, the following estimates hold: with (still formally) c i = ∂ ξ k c i , and the orthogonality conditions now become We consider the constants b i defined as and the functions By (2.30), (3.3) and (3.4), a straightforward but tedious computation shows that ∂ ξ k W * ,∂Ωt = O (1), Furthermore, using (2.18), (2.28), (3.6), (3.29), (3.30) and the fact that 1 γ k ≤ C uniformly on t, we find a * * ,Ωt ≤ Ct h * ,∂Ωt and b * ,∂Ωt ≤ Ct h * ,∂Ωt .
We then have By the remark above it follows that this equation has a unique solution Z, c 1 , . . . , c m , and thus This, together with the fact that χ i Z 1i L ∞ (Ωt) = O (1/γ i ), implies that (3.88) holds.

The nonlinear problem.
In what follows we shall consider first the intermediate problem: for any points ξ = (ξ 1 , . . . , ξ m ) ∈ O t , we find a function φ, and scalars c 1 , . . . , c m , such that

From Proposition 3.1, we get
Hence we get where C is independent of κ. Then by (2.8), (2.11), (2.18) and (2.28), it follows that for all t sufficiently large A is a contraction on F κ , and therefore a unique fixed point of A exists in this region. Let us now analyze the differentiability of φ. Since R depends continuously (in the *-norm) on the m-tuple ξ = (ξ 1 , . . . , ξ m ), the fixed point characterization obviously yields so for the map ξ → φ. Assume for instance that the partial derivative ∂ ξ k φ exists. Since φ = T (N (φ) + R), formally that From Lemma 3.4 we deduce that Also observe that we have so that, using the fact that ∂ ξ k W * ,∂Ωt = O (1), Since ∂ ξ k R * ,∂Ωt = O t β max i {εγ i } , and by Proposition 3.1, we conclude from the above computation that The above computation can be made rigorous by using the implicit function theorem and the fixed point representation (4.4) which guarantees C 1 regularity of ξ . This problem is indeed variational: it is equivalent to finding critical points of a function of ξ = εξ . To realize it we consider the energy function J t associated to problem (1.4), namely where U is the function defined in (2.12) andφ(ξ)(x) = φ( x ε , ξ ε ), x ∈ Ω, with φ = φ ξ the unique solution to problem (4.1) given by Proposition 4.1. Critical points of F t correspond to solutions of (5.1) for large t, as the following result states.

Proof.
A direct consequence of the results obtained in Proposition 4.1 and the definition of function U (ξ) is the fact that the map ξ → F t (ξ) is of class C 1 . Define Then, making a change of variables, we have Let us assume that D ξ F t (ξ) = 0. Then for all k = 1, . . . , m, This implies that system (5.6) is diagonal dominant, and thus c i = 0 for all i = 1, . . . , m.

BOUNDARY BUBBLING SOLUTIONS 5497
We also have the validity of the following lemma.
Proof. First, from (5.4)-(5.5), we have Taking into account DI t (V + φ ξ )[φ ξ ] = 0, a Taylor expansion and an integration by parts give Let us differentiate the representation (5.8) with respect to ξ k , k = 1, . . . , m, The continuity in ξ of all these expressions is inherited from that of φ and its derivatives in ξ in the L ∞ norm.
6. Expansion of energy and proof of Theorem 1.1. In this section we write an asymptotic expansion of the energy functional J t evaluated at U (ξ) and further give the proof of Theorem 1.1.
We have Proposition 6.1. Let m be a positive integer. With the choice (2.17) for the parameters µ i , there exists t m > 1 such that for any t > t m and any points ξ = (ξ 1 , . . . , ξ m ) ∈ O t , the following expansion holds uniformly:  As a consequence, from (6.2) and (6.3) we can write an asymptotic expansion of the energy (5.2) evaluated at U (ξ), namely

Making the change of variables
This, together with the definition (2.11) of ε i and the choice (2.17) of µ i , implies that (6.1) holds.
We now have all the ingredients needed to give the proof of Theorem 1.1.