Let Ω be a bounded domain in with smooth boundary and . This paper is concerned with the existence of positive solutions to the boundary value problem
where ν denotes the outer unitary normal vector of . Elliptic equations with nonlinear Neumann boundary condition of exponential type arise in conformal geometry (prescribing Gaussian curvature of the domain and curvature of the boundary), see for instance [9, 10, 24] and references therein, and in corrosion modelling, see [7, 20, 26, 27].
Problem (1.1) is the Euler–Lagrange equation for the functional
This optimal embedding is related to the critical Trudinger–Moser trace inequality
see . It has been proven  that for any bounded domain Ω in , with smooth boundary, the supremum is attained by a function with . Furthermore, for any , the supremum is finite and it is attained, while as soon as . See also [11, 21, 22, 25, 30] for generalizations. Observe that critical points of the above constrained variational problem satisfy, after a simple scaling, an equation of the form (1.1).
The Trudinger–Moser trace embedding is critical, involving loss of compactness analogous to that related to the Trudinger–Moser embedding for functions u with zero boundary value,
for which the analogous problem to (1.1) is
whose energy functional is given by ,
It is known that satisfies the compactness PS-condition for energy levels less that (see ). Loss of compactness in is described by the presence of families of blowing-up solutions for problem (1.2). It has been proven in  that if solves problem (1.2) for , with bounded and , then, passing to a subsequence, there is an integer such that
see [2, 19, 5]. This quantization property is not known for general Palais–Smale sequences associated to (see ). When , a more precise description of the blowing-up behavior of these families of solutions is known . On the other hand, a simple observation is that the functional has a mountain pass geometrical structure. In fact, in [1, 6] it is shown that there exists such that for all , the mountain pass level stands below where the PS-condition holds. Thus a solution to (1.2) always exists for this range of values of λ. As , the family of mountain pass solutions satisfies (1.3) with . In  it is proven that if Ω has a sufficiently small hole, a solution to (1.2), satisfying (1.3), exists. Further results were obtained in : if Ω has a hole of any size, namely Ω is not simply connected, then a solution satisfying property (1.3) with exists. This solution happens to blow up exactly at two points in Ω. General conditions for the existence of solutions of problem (1.2) for small λ, which satisfy the bubbling condition (1.3), for any , are provided in , together with the precise characterization of their blow-up profile. In fact, blowing-up solutions satisfying (1.3) happen to blow up at exactly k points which are located in the interior of Ω. See also [4, 14, 16] for related results.
In this paper, we are concerned with the construction of solutions to (1.1), in the same spirit as the result described above in . Assume that Ω is any bounded domain with smooth boundary. For any integer k we find existence of a pair of solutions to problem (1.1) for small λ, whose energy satisfy the bubbling condition
Furthermore, we give a precise description of their bubbling behavior.
To state our result, let us introduce the function with
where G is the Green function for the Neumann problem
and H its regular part defined as
In this paper we establish the following:
Let Ω be a bounded domain in with smooth boundary and let be an integer. Then there exists such that, for all small λ with , there exists a pair of solutions , to problem (1.1) such that
where as . Moreover, for any , passing to a subsequence, there exists
with , such that and
where on each compact subset of .
These solutions blow up at points located near , while far away from these points the solutions look like a combination of Green function with positive weights . These points and parameters correspond to two distinct critical points of .
We can actually show a stronger version of this result. If has more than one component, then pairs of families of solutions blowing up at k points on each component happen to exist. In reality, associated to each topologically nontrivial critical point situation associated to (for instance local maxima or saddle points possibly degenerate), a solution with concentration peaks at a corresponding critical point exists. We will not elaborate more on this point, and we refer the interested reader to .
It is important to remark the interesting analogy between these results and those known for other problems with exponential nonlinearity on the boundary, as
In , a construction of solutions to (1.8) with bounded is carried out: for any integer , there are at least two distinct families of solutions which approach the sum of k Dirac masses at the boundary. The location of these possible points of concentration may be further characterized as critical points of the functional of k points of the boundary defined as
where G and H are defined in (1.5) and (1.6), respectively. Observe that the function only depends on the points on the boundary and it does not depend on positive parameters . This is completely different from the case of the function which is defined in (1.4) and which determines the bubbling behavior of solutions to (1.1). Furthermore, it has been proven that, far from , the solutions to problem (1.8) found in  look like
Thus, also the solutions to problem (1.8) found in  are combinations of Green function, far from the concentration points, but unlike the solutions obtained in Theorem 1.1 for problem (1.1), the weights in front of the Green functions are always equal to 1. Thus, to construct solutions to problem (1.1), not only we have to find the location of the bubbling points on the boundary, but also the weights in front of the Green functions in (1.7).
The solutions predicted in Theorem 1.1 are constructed as a small additive perturbation of an appropriate initial approximation. A linearization procedure leads to a finite-dimensional reduction, where the reduced problem corresponds to that of adjusting variationally the location of the concentration points and of the weights . A precise description of the approximation and a detailed outline of the proof and of the organization of the paper are given in Section 2.
Let us just mention that through out the paper, C will always denote an arbitrary positive constant, independent of λ, whose value changes from line to line.
2 A first approximation and outline of the argument
It is useful for our purpose to consider the change of variables so that problem (1.1) gets rewritten as
The first part of this section is devoted to constructing a good approximation for a solution to problem (2.1) and to estimate its error. To do so, let us introduce the following problem in the entire plane
We next describe an approximate solution to (2.1) whose shape is given by the sum of functions centered at points on the boundary of Ω and properly scaled. Let k be an integer, let be points on the boundary of Ω and let be positive numbers. We assume there exists a positive, small number δ such that
We thus define the functions
where is the unique solution to the problem
In the above definitions, and are positive numbers. These numbers and will be defined later on in terms of and in order to ensure that is a function very close to a solution for problem (2.1). Let us just mention that, a posteriori, the parameters will tend to zero, as , namely
while the numbers will remain bounded from above and strictly positive, as . Taking this into account, we easily see that the shape of the function change depending whether you evaluate it far from the fixed points or in a region very close to one of the points . Let us then describe carefully the shape of in these two regions. For this purpose, we need the following:
where denotes a function in Ω which is uniformly bounded as , and H is the regular part of Green’s function defined in (1.5).
We refer the reader to  for a proof of this lemma. ∎
A direct consequence of Lemma 2.1 is that, for a given small and fixed, in the region for all , the function looks like
Here and in what follows, with we denote a general function in Ω of the form , where is uniformly bounded in Ω as .
Let us now examine in a neighborhood of a given . Assume that and set , . Explicit computations give that
as . We set
We thus write the above expansion in the following compact form: for ,
The solution to (2.1) we are looking for the form
where is defined as in (2.5), and ϕ represents a lower order correction. In fact, we aim at finding a solution for a function ϕ small in some proper sense provided that the points and the parameters are suitably chosen. Assuming for the moment that ϕ is small, we rewrite problem (2.1) as follows:
Here and in what follows f denotes the nonlinearity
It is not hard to believe that having a good approximation to a solution of problem (2.1) is reflected into the fact that the function E is small, in some sense to be made precise. It is in this context that we will choose and in such a way that the error of approximation E for is small around each point under some appropriate norm.
Let us be more precise. The error E is clearly defined by (2.12). Assume that is a small but fixed positive number and with . In this region, we have that
as . We thus choose to be defined as
On the other hand, in the same region, we have
Thus, in order to match at main order the two terms and in a region near the point , we fix the parameter such that the number satisfies
This condition defines the parameter as follows:
With these choices of we get
As a conclusion, the choice we made of and of gives that in the region , the error of approximation can be described as follows:
Let us mention now that a direct computation shows that for , ,
Then we get
for some positive constant C. Set . Let be the characteristic function on and
Taking into account (2.14), we get the following global bound on the error of approximation:
We define the -weight norm
We thus have the validity of the following key estimate for the error term E:
We conclude this section explaining the strategy to solve problem (2.11), which guarantees the existence of a solution to problem (2.1) of the form (2.10). In fact, we will solve problem (2.11) in two steps. The first step consists in solving problem (2.11) in a projected space. Let us be more precise.
Define in ,
For , we define to be a diffeomorphism, where M is an open neighborhood of the origin in such that , . We can select so that it preserves area. Define
Next, let us consider a large but fixed number and a nonnegative radial and smooth cut-off function χ with if and if , . Then set
The problem we first solve is to find a function ϕ and numbers such that
Consider the norm
We prove the following:
Let be a small but fixed number. Assume that the points and the parameters satisfy
Then there exist positive numbers and C, such that, for any , problem (2.21) has a unique solution ϕ, which satisfies
for all . Moreover, if we consider the map into the space , the derivative and exists and defines a continuous function of . Besides, there is a constant such that
for all .
The proof of this result is contained in Section 3.
At this stage of our argument, we have solved the nonlinear problem (2.21). In order to find a solution to the original problem we need to find ξ and m such that
This problem is indeed variational: it is equivalent to finding critical points of a function of ξ and m. Associated to (1.1), let us consider the energy functional given by
and the finite-dimensional restriction
The proof of the above proposition, together with the expansion of the functional is given in Section 4. Section 5 is devoted to concluding the proof of Theorem 1.1. The final Appendix, Section 6, contains the proofs of some estimates we have used through the paper.
3 Proof of Proposition 2.1
The proof of Proposition 2.1 is based on a fixed point argument and the invertibility property of the following linear problem: Given , find a function ϕ and constants such that
We shall prove the validity of the following proposition:
Let be a small but fixed number and assume that we have and with
Then there exist positive numbers and C such that, for any and any , there is a unique solution , and to (3.1). Moreover,
The proof of this result is based on the a-priori estimate for solutions to the following problem:
where . We have the validity of the following lemma.
We will carry out the proof of the a priori estimate (3.4) by contradiction. We assume then the existence of sequences , points and numbers , which satisfy relations (3.2) and (2.16), functions , with , with , constants ,
We will prove that in reality under the above assumption we must have that uniformly in , which is a contradiction that concludes the result of the lemma. We will divide into the following several steps to prove this.
Passing to a subsequence we may assume that the points approach limiting, distinct points in . Indeed, let us observe that locally uniformly in , away from the points . Away from the points we have then uniformly on compact subsets on . Since is bounded, it follows also that passing to a further subsequence, approaches in local sense on compacts of a limit which is bounded and satisfies in . Furthermore, observe that far from , locally uniformly on and so we also have on . Hence extends smoothly to a function which satisfies in Ω, and on . We conclude that , and the claim follows.
Indeed, for notational convenience, we shall omit the explicit dependence on n in the rest of the proof. Multiplying the first equation of (3.3) by and integrating over , we find
Having in mind that in sense in , we have
Furthermore, a direct computation shows that
where is some universal constant and if , and if . On the other hand, we have
In fact, estimate (3.8) is a direct consequence of the definition of the -norm. Let us prove the validity of (3.7). Recall that in , we have that , where is chosen to preserve area (see (2.19)). Performing the change of variables , we get that
where and is a second order differential operator defined as follows:
On the other hand, we observe that, after a possible rotation, we can assume that . Hence, using again the change of variables , we get
where with , and is a positive function, coming from the change of variables, which is uniformly positive and bounded as . Furthermore, B is a differential operator of order one on . In fact, we have
On the other hand, since
for some . Thus we can conclude that
This shows the validity of (3.7).
We shall now estimate the term . Using the definition of the -norm, we observe that
Since are uniformly bounded, as , in , we just need to estimate
Recall that the functions are defined as with , , and . Using the change of variables , we have
where and Therefore we get
Set , and . Arguing as in [12, Lemma 4.3], we have that for small enough, there exist , and
smooth, positive and bounded function such that
Thus, by maximum principle in , the function can be bounded by
for some constant independent of λ. Thus we get (3.14).
By the maximum principle and the Hopf Lemma we find that
Thus, we can find that there is some fixed such that
Set , and consider the change of variables
Then by elliptic estimate (up to subsequence) converges uniformly on compact sets to a nontrivial solution of the problem
By the nondegeneracy result , we conclude that is a linear combination of and . On the other hand, we can take the limit in the orthogonality relation and we find that for . This contradicts the fact that . This ends the proof of the lemma. ∎
Proof of Proposition 3.1.
In proving the solvability of (3.1), we may first solve the following problem: For given , with bounded, find and , , , such that
First we prove that for any ϕ, solution to (3.15) the bound
holds. In fact, by Lemma 3.1, we have
and therefore it is enough to prove that .
Fix an integer j. To show that , we shall multiply equation (3.15) against a test function, properly chosen. Let us observe that the proper test function depends whether we are considering the case or . We start with . We define , where
In fact, we recognize that in , on and on . Let and be two smooth cut-off functions defined in as
We assume that (see (2.20)) and we define
for . We multiply equation (3.15) against and we integrate by parts. We get
Observe first that, assuming , we have
Furthermore, we have
We claim that
Observe first that, assuming , we have
Using the change of variables , we get
where and . But and so
where and is a positive function, coming from the change of variables, which is uniformly positive and bounded as . Observe that
for and , and this implies that
for some . Thus we can conclude that
where . We thus compute in , with ,
On the other hand, in this region we have . Thus
Summarizing all the above information, we get
endowed the norm . Problem (3.15), expressed in a weak form, is equivalent to find such that
Then there is a unique solution of (3.24), and
for some constant C independent on λ.
Multiplying (3.24) by , and integrating by parts, we have
where are Kronecker’s delta. Then we get
with independent of . Hence the matrix (or ) with entries (or ) in invertible for small and () uniformly in .
Now, given we find , , solution to (3.15). Define constants as
The above linear system is almost diagonal, since arguing as before one can show that
where is a positive universal constant. Then define
A direct computation shows that ϕ satisfies (3.1) and furthermore
A slight modification of the proof above also shows that for any and , with , , the problem
has a unique solution ϕ, , , , satisfying
with C independent of λ.
The operator is differentiable with respect to the variable on satisfying (3.2), and for , one has the estimate
for a given positive C, independent of λ, and for all λ small enough.
Differentiating equation (3.1), formally for all , should satisfy in Ω the equation
and on the boundary ,
with , and the orthogonality conditions now become
We consider the constants , , , defined as
We then have
Hence, using the result of Proposition 3.1 we have . By the definition of , we get . Since and , we obtain Hence we get
An analogous computation holds true if we differentiate with respect to . ∎
We are now in a position to prove Proposition 2.1.
Proof of Proposition 2.1.
For a given number , let us consider the region
From Proposition 3.1, we get . We claim that
We then get that for a sufficiently large but fixed γ and all small λ. Moreover, for any , a straightforward computation gives
Thus we have
Hence the operator A has a small Lipschitz constant in for all small λ, and therefore a unique fixed point of A exists in this region.
We shall next analyze the differentiability of the map . Assume for instance that the partial derivative exists for , . Since , formally we have that
From (3.25), we have . On the other hand,
Since , Proposition 3.1 guarantees that for all . An analogous computation holds true if we differentiate with respect to . Then the regularity of the map can be proved by standard arguments involving the Implicit Function Theorem and the fixed point representation (3.26). This concludes proof of the proposition. ∎
4 Proof of Proposition 2.2 and expansion of the energy
Up to now we have solved the nonlinear problem (2.21). In order to find a solution to the original problem, we need to find ξ and m such that
We recall the following definitions: the energy functional associated to problem (1.1) is
and the finite-dimensional restriction
where ϕ is the unique solution to problem (2.21) given by Proposition 2.1. Critical points of correspond to solutions of (4.1) for a small λ, as the result of Proposition 2.2 states. We give the proof of this result.
Proof of Proposition 2.2.
A direct consequence of the results obtained in Proposition 2.1 and the definition of function is the fact that the map is of class .
From Proposition 2.1, we observe that
We can rewrite
Since is the solution of (2.21), it follows that satisfies
where . For any l, we define
We note that We compute the differential , ; thus we have
Now, fix i and j. We compute the coefficient in front of . To this end, we choose and obtain
Thus we concludes that for any , we have
Similarly, we get that for all ,
Thus, we can conclude that is equivalent to the following system:
for some fixed constant A, with small in the sense of the -norm as . The conclusion of the lemma follows if we show that the matrix of dimension is invertible in the range of the points and parameters we are considering. Indeed, this fact implies unique solvability of (4.2). Inserting this in (4.3), we get unique solvability of (4.3).
Consider the definition of the , in terms of the parameters and points given in (3.2). These relations correspond to the gradient of the function defined as
We set . Then the above function can be written as
This function is a strictly convex function of the parameters , for parameters uniformly bounded and uniformly bounded away from 0 and for points in Ω uniformly far away from each other and from the boundary. For this reason, the matrix is invertible in the range of parameters and points we are considering. Thus, by the Implicit Function Theorem, relation (2.16) defines a diffeomorphism between and . This fact gives the invertibility of . This concludes the proof of the lemma. ∎
In order to solve for critical points of the functional , a key step is its expected closeness to the functional . This fact is contained in the following lemma.
The following expansion holds:
uniformly on points and parameters satisfying the constraints in Proposition 3.1.
Taking into account , a Taylor expansion gives
Since , we have . Let us differentiate with respect to ξ. We use the representation (4.4) and differentiate directly under the integral sign; we get that, for all ,
Since and by the computations in the proof of Lemma 2.1, we get
With the same argument, we get
The continuity in ξ and m of all these expressions is inherited from that of ϕ and its derivatives in ξ and m in the -norm. This concludes the proof. ∎
We end this section with the asymptotic estimate of , where
and is the energy functional associated to (1.1), whose definition is as follows:
We have the following result.
Let be given by (2.16). Then
where denotes the measure of , and is a function, uniformly bounded with its derivatives, as , for points ξ and parameters m satisfying (3.2). Furthermore, the function
is defined by
Let us set with , where and is defined in (2.6). Then
First, we write
Multiplying (2.6) by , it yields
and multiplying (2.6) by again, we find
Then we get
Taking the change of variables , we have
Using the definition of , we thus conclude that
On the other hand, we have
Multiplying (2.6) by and integrating, we find
Multiplying (2.6) by again and integrating, we find
Then we conclude that
Finally, let us evaluate the third term in the energy
with a function, uniformly bounded with its derivatives, as . And
By the choice of in (2.16), we get that the function in the expansion (4.5) is uniformly bounded, as , for points ξ and parameters m satisfying (3.2). In order to prove that also the derivatives, in ξ and in m, of this function are uniformly bounded, as , in the same region, one argues similarly as for the expansion of . We leave the details to the reader. Thus the proof of this lemma is complete. ∎
5 Proof of Theorem 1.1
In this section, we will prove the main result.
Proof of Theorem 1.1.
Let be the open set such that
From Proposition 2.2, the function
where defined by (2.5) and ϕ is the unique solution to problem (2.21) given by Proposition 2.1, is a solution of problem (1.1) if we adjust so that it is a critical point of defined by (2.23). This is equivalent to finding a critical point of
where and are uniformly bounded in consider region as . Thus we need to find a critical point of
We make the change of variables , and set . And we next find a critical point of
which is well defined on . For , we have
We have that is strictly convex as a function of s, and it is bounded below. Hence it has a unique minimum point, which we denote by , each component of is a function of points , namely , satisfying
is a function with respect to ξ defined in .
There is a positive constant such that for each .
as for some .
In fact, (1) directly holds by the Implicit Function Theorem. Moreover, since is positive and is bounded, from (5.2) we have
Then we get (2). Furthermore, for some , we have as , so (3) holds by (5.2).
A direct computation shows that
Given one component of , let be a continuous bijective function that parameterizes . Set
Next, we find a critical point of . The function is , bounded from above in , and from (3) we have
Hence, since δ is arbitrarily small, has an absolute maximum M in .
On the other hand, using Ljusternik–Schnirelmann theory as in the proof in , we get that has at least two distinct points in . Let be the Ljusternik–Schnirelmann category of relative to , which is the minimum number of closed and contractible in sets whose union covers . We will estimate the number of critical points for below by .
Indeed, by contradiction, suppose that . This means that is contractible in itself, namely there exist a point and a continuous function such that, for all ,
Define to be the continuous function given by
Let be the well-defined continuous map given by
where is the projection on the first component. The function η is a contraction of to a point and this gives a contradiction; then the claim follows.
Therefore we have for any . Define
Then by the Ljusternik–Schnirelmann theory we obtain that c is a critical level.
If , we conclude that has at least two distinct critical points in . If , there is at least one set C such that , where the function reaches its absolute maximum. In this case we conclude that there are infinitely many critical points for in .
Thus we obtain that the function has at least two distinct critical points in , denoted by . Hence and are two distinct critical points for the function . From (5.1) we then have that has at least two critical points. This ends the proof of Theorem 1.1. ∎
Proof of the first estimate in (3.21).
We shall prove
where is defined in (3.18). Perform the change of variables and denote
where is defined in (3.10). We shall show that
This fact implies the first estimate in (3.21).
Let us first consider the region where . In this region, . Since and since (3.10) holds, we have
In the region , we have . Therefore, in this region,
For the other terms we find
In the region the definition of is
We will estimate each term of (3.10) using the facts that , and that in the considered region which implies also . We obtain, for ,
Similarly, for , we have
This shows that
Thus we only need to estimate the size of in the region . In this region we have
First we recall that and, for ,
Proof of the second estimate in (3.21).
We shall prove
We perform the change of variables . We already observed that we can assume that . Hence,
Next, in the region we have
Since h is radial, this implies
Using (3.13), we see that
Using the fact that h has zero normal derivative on , we deduce
On the other hand, using (3.13), we have in that
for some . Finally, we consider . In this region we have and , . Using these facts, estimate (6.6) and that has zero normal derivative, we find
for . From (3.13) we have
Thus we conclude that for , ,
Proof of the first estimate in (3.27).
We shall prove
For , far away from the points , namely for , i.e. for all , a consequence of (2.8) is that , . Then we have
where is the characteristic function of the set . Moreover, for , we have
Then we find
Proof of the second estimate in (3.27).
Thus we obtain
Collecting all this information and using the definition of the -norm, we obtain the second estimate in (3.27). ∎
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About the article
Published Online: 2017-07-28
The research of the first author has partly been supported by NSFC No. 11501469 and Chongqing Research Program of Basic Research and Frontier Technology cstc2016jcyjA0032. The research of the second author has partly been supported by Fondecyt Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 615–644, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0092.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0