Open Access

Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

Boundary Value Problems20112011:10

DOI: 10.1186/1687-2770-2011-10

Received: 22 March 2011

Accepted: 12 August 2011

Published: 12 August 2011

Abstract

Given Ω bounded open regular set of 2 and x1, x2, ..., x m Ω, we give a sufficient condition for the problem

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equa_HTML.gif

to have a positive weak solution in Ω with u = 0 on ∂Ω, which is singular at each x i as the parameters ρ, λ > 0 tend to 0 and where f(u) is dominated exponential nonlinearities functions.

2000 Mathematics Subject Classification: 35J60; 53C21; 58J05.

Keywords

singular limits Green's function nonlinear Cauchy-data matching method

1 Introduction and statement of the results

We consider the following problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ1_HTML.gif
(1)
where is the gradient and Ω is an open smooth bounded subset of 2. The function a is assumed to be positive and smooth. In the following, we take a(u) = e λu and f(u) = e λu (e u + e γu ), for λ > 0 and γ (0, 1), then problem (1) take the form
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ2_HTML.gif
(2)
Using the following transformation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equb_HTML.gif
then the function w satisfies the following problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ3_HTML.gif
(3)

with ϱ = (λρ2)1-λ. So when λ → 0+, the exponent https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq1_HTML.gif tends to infinity while the exponent https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq2_HTML.gif tends to -∞. For ϱ ≡ 0, problem (3) has been studied by Ren and Wei in [1]. See also [2].

We denote by ε the smallest positive parameter satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ4_HTML.gif
(4)
Remark that ρ ~ ε as ε → 0. We will suppose in the following
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equc_HTML.gif
In particular, if we take https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq3_HTML.gif , then the condition (A λ ) is satisfied. Under the assumption (A λ ), we can treat equation (2) as a perturbation of the following:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equd_HTML.gif

for γ (0, 1).

Our question is: Does there exist vε,λa sequence of solutions of (2) which converges to some singular function as the parameters ε and λ tend to 0?

In [3], Baraket et al. gave a positive answer to the above question for the following problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ5_HTML.gif
(5)

with a regular bounded domain Ω of 2. They give a sufficient condition for the problem (5) to have a weak solution in Ω which is singular at some points (x i )1≤imas ρ and λ a small parameters satisfying (A λ ), where the presence of the gradient term seems to have significant influence on the existence of such solutions, as well as on their asymptotic behavior.

In case λ = 0 the authors in [4] gave also a positive answer for the following problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ6_HTML.gif
(6)
for γ (0, 1) as ρ tends to 0. When λ = 0 and γ = 0, problem (2) reduce to
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ7_HTML.gif
(7)

The study of this problem goes back to 1853 when Liouville derived a representation formula for all solutions of (7) which are defined in 2, see [5]. It turns out that, beside the applications in geometry, elliptic equations with exponential nonlinearity also arise in modeling many physical phenomenon, such as thermionic emission, isothermal gas sphere, gas combustion, and gauge theory [6]. When ρ tends to 0, the asymptotic behavior of nontrivial branches of solutions of (7) is well understood thanks to the pioneer work of Suzuki [7] which characterizes the possible limit of nontrivial branches of solutions of (7). His result has been generalized in [8] to (6) with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq4_HTML.gif , and finally by Ye in [9] to any exponentially dominated nonlinearity f(u). The existence of nontrivial branches of solutions with single singularity was first proved by Weston [10] and then a general result has been obtained by Baraket and Pacard [11]. These results were also extended, applying to the Chern-Simons vortex theory in mind, by Esposito et al. [12] and Del Pino et al. [13] to handle equations of the form -Δu = ρ2V(x)e u where V is a nonconstant positive potential. See also [1416] wherever this rule is applicable. where the Laplacian is replaced by a more general divergence operator and some new phenomena occur. Let us also mention that the construction of nontrivial branches of solutions of semilinear equations with exponential nonlinearities allowed Wente to provide counter examples to a conjecture of Hopf [17] concerning the existence of compact (immersed) constant mean curvature surfaces in Euclidean space. Another related problem is the higher dimension problem with exponential nonlinearity. For example, the 4-dimensional semilinear elliptic problem with bi-Laplacian is treated in [18] and the problem with an additional singular source term given by Dirac masses is treated in [19] in the radial case. The results in [18, 19] are generalized to noncritical points of the reduced function, see [20].

We introduce now the Green's function G(x, x') defined on Ω × Ω, to be solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Eque_HTML.gif
and let H(x, x') = G(x, x') + 4log |x - x'|, its regular part. Let m , we set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ8_HTML.gif
(8)

which is well defined in (Ω) m for x i x j for ij. Our main result is the following

Theorem 1 Given β (0, 1). Let Ω an open smooth bounded set of 2, λ > 0 satisfying the condition (A λ ), γ (0, 1) and S = {x1, ... x m } Ω be a nonempty set. Assume that, the point (x1, ..., x m ) is a nondegenerate critical point of the function
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equf_HTML.gif
then there exist ε0 > 0, λ0 > 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq5_HTML.gif a family of solutions of (2), such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equg_HTML.gif

One of the purpose of the present paper is to present a rather efficient method: nonlinear Cauchy-data matching method to solve such singularly problems. This method has already been used successfully in geometric context (constant mean curvature surfaces, constant scalar curvature metrics, extremal Kähler metrics, manifolds with special holonomy, ...) and appeared in the study [18] in the context of partial differential equations.

2 Construction of the approximate solution

We first describe the rotationally symmetric approximate solutions of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ9_HTML.gif
(9)
in 2 which will play a central role in our analysis. Given ε > 0, we define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ10_HTML.gif
(10)
which is clearly a solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ11_HTML.gif
(11)
in 2. Let us notice that equations (11) is invariant under dilation in the following sense: If v is a solution of (11) and if τ > 0, then v(τ ·) + 2logτ is also a solution of (11). With this observation in mind, we define for all τ > 0
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ12_HTML.gif
(12)

2.1 A linearized operator on 2

For all ε, τ, λ > 0, we set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ13_HTML.gif
(13)
for δ (0, 1). We define the linear second order elliptic operator
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ14_HTML.gif
(14)
which corresponds to the linearization of (11) about the solution u1 (= uε = τ = 1) given by (10) which has been defined in the previous section. We are interested in the classification of bounded solutions of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif in 2. Some solutions are easy to find. For example, we can define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equh_HTML.gif
where r = |x|. Clearly https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq7_HTML.gif and this reflects the fact that (11) is invariant under the group of dilations τu(τ ·) + 2 logτ. We also define, for i = 1, 2
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equi_HTML.gif

which are also solutions of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq8_HTML.gif . Since, these solutions correspond to the invariance of the equation under the group of translations au(· + a). We recall the following result which classifies all bounded solutions of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif which are defined in 2.

Lemma 1 [11] Any bounded solution of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif defined in 2 is a linear combination of ϕ i for i = 0, 1, 2.

Let B r denote the ball of radius r centered at the origin in 2.

Definition 1 Given k , β (0, 1) and μ , we introduce the Hölder weighted spaces https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq9_HTML.gif as the space of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq10_HTML.gif for which the following norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equj_HTML.gif

is finite.

We define also
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equk_HTML.gif

As a consequence of the result of Lemma 1, we recall the surjectivity result of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq11_HTML.gif given in [11].

Proposition 1 [11]
  1. (i)
    Assume that μ > 1 and μ , then
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equl_HTML.gif
     
is surjective.
  1. (ii)
    Assume that δ > 0 and δ then
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equm_HTML.gif
     

is surjective.

We set https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq12_HTML.gif , we define

Definition 2 Given k , β (0, 1) and μ , we introduce the Hölder weighted spaces https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq13_HTML.gif as the space of functions in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq14_HTML.gif for which the following norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equn_HTML.gif

is finite.

Then, we define the subspace of radial functions in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq15_HTML.gif by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equo_HTML.gif
We would like to find a solution u of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ15_HTML.gif
(15)
in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq16_HTML.gif . By using the transformation, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq17_HTML.gif then Eq. (15) is equivalent to
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ16_HTML.gif
(16)
in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We look for a solution of (16) of the form v(x) = u1(x) + h(x), this amounts to solve
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ17_HTML.gif
(17)

In https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We will need the following:

Definition 3 Given https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq19_HTML.gif , k, β (0, 1) and μ , the weighted space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq20_HTML.gif is defined to be the space of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq21_HTML.gif endowed with the norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equp_HTML.gif
For all σ ≥ 1, we denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq22_HTML.gif the extension operator defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ18_HTML.gif
(18)
where t α χ(t) is a smooth non-negative cutoff function identically equal to 1 for t ≤ 1 and identically equal to 0 for t ≥ 2. It is easy to check that there exists a constant c = c(μ) > 0, independent of σ ≥ 1, such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ19_HTML.gif
(19)
We fix δ (0, 1) and denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq23_HTML.gif to be a right inverse of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq24_HTML.gif provided by Proposition 1. To find a solution of (17), it is enough to find a fixed point h, in a small ball of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq25_HTML.gif , solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ20_HTML.gif
(20)
We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equq_HTML.gif
This implies that given κ > 0, there exist c κ > 0 (only depend on κ), such that for δ (0,1) and |x| = r, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equr_HTML.gif
Making use of Proposition 1 together with (19), we conclude that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ21_HTML.gif
(21)
Now, let h1, h2 such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq26_HTML.gif in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq25_HTML.gif , then for δ (0, 1 - r] we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equs_HTML.gif
Similarly, making use of Proposition 1 together with condition (A λ ) and (19), we conclude that given κ > 0, there exist ε κ > 0, λ κ > 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq27_HTML.gif (only depend on κ) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ22_HTML.gif
(22)

Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq28_HTML.gif for all λ (0, λ κ ) and ε (0, ε κ ). Then, (21) and (22) are enough to show that h is a contraction from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq29_HTML.gif into itself and hence has a unique fixed point h in this set. This fixed point is solution of (20) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We summarize this in the:

Proposition 2 Given δ (0, 1 - γ] and κ > 1, then there exist https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (independent of ε and λ) and a unique https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq31_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq32_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equt_HTML.gif

solves (16) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif .

2.2 Analysis of the Laplace operator in weighted spaces

In this section, we study the mapping properties of the Laplace operator in weighted Hölder spaces. Given x1, ..., x m Ω, we define x := (x1, ..., x m )
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equu_HTML.gif
and we choose r0 > 0 so that the balls https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq33_HTML.gif of center x i and radius r0 are mutually disjoint and included in Ω. For all r (0, r0), we define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equv_HTML.gif

With these notations, we have:

Definition 4 Given k , β (0,1) and ν , we introduce the Hölder weighted space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq34_HTML.gif as the space of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq35_HTML.gif for with the following norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equw_HTML.gif

is finite.

When k ≥ 2, we denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq36_HTML.gif be the subspace of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq37_HTML.gif satisfying w = 0 on ∂Ω. We recall the

Proposition 3 [21] Assume that ν < 0 and ν , then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equx_HTML.gif

is surjective. Denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq38_HTML.gif a right inverse of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq39_HTML.gif .

Remark 1 Observe that, when ν < 0, ν , the right inverse even though is not unique and can be chosen to depend smoothly on the points x1, ..., x m , at least locally. Once a right inverse is fixed for some choice of the points x1, ..., x m , a right inverse which depends smoothly on some points close to x1, ..., x m can be obtained using a simple perturbation argument. This argument will be used later in the nonlinear exterior problem, since we will move a little bit the points (x i ).

2.3 Harmonic extensions

We study the properties of interior and exterior harmonic extensions. Given https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq41_HTML.gif and define H i (=H i (φ; ·)) to be the solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ23_HTML.gif
(23)

We denote by e1, e2 the coordinate functions on S1.

Lemma 2 [21] If we assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ24_HTML.gif
(24)
then there exists c > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equy_HTML.gif
Given https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq42_HTML.gif , we define https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq43_HTML.gif to be the solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ25_HTML.gif
(25)

which decays at infinity.

Definition 5 Given k , β (0,1) and ν , we define the space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq44_HTML.gif as the space of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq45_HTML.gif for which the following norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equz_HTML.gif

is finite.

Lemma 3 [21] If we assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ26_HTML.gif
(26)
Then there exists c > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaa_HTML.gif

If F L2(S1) is a space of functions defined on S1, we define the space F to be the subspace of functions F of which are L2(S1) -orthogonal to the functions 1, e1,e2. We will need the:

Lemma 4 [21] The mapping
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equab_HTML.gif

where H i (= H i (ψ; ·)) and H e = H e (ψ; ·), is an isomorphism.

3 The nonlinear interior problem

We are interested in studying equations of type
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ27_HTML.gif
(27)

In https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif .

Given https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq46_HTML.gif satisfying (24), we define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equac_HTML.gif
Then, we look for a solution of (27) of the form w = v + v and using the fact that H i is harmonic, this amounts to solve
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ28_HTML.gif
(28)
We fix μ (1,2) and denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq47_HTML.gif to be a right inverse of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq48_HTML.gif provided by Proposition 1. To find a solution of (28), it is sufficient to find https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq49_HTML.gif solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ29_HTML.gif
(29)

We denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq50_HTML.gif , the nonlinear operator appearing on the right-hand side of equation (29).

Given κ > 0 (whose value will be fixed later on), we further assume that the functions φ satisfy
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ30_HTML.gif
(30)

Then, we have the following result

Lemma 5 Given κ > 0. There exist ε κ > 0, λ κ > 0, c κ > 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (only depend on κ) such that for all λ (0, λ κ ) and ε (0, ε κ )
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equad_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equae_HTML.gif

provided https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq51_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq52_HTML.gif .

Proof. The proof of the first estimate follows from the asymptotic behavior of H i together with the assumption on the norm of boundary data φ given by (30). Indeed, let c κ be a constant depending only on κ (provided ε and λ are chosen small enough) it follows from the estimate of H i , given by lemma 2, that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaf_HTML.gif
Since for each https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq53_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equag_HTML.gif
where δ (0, 1 - γ]. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equah_HTML.gif
On the other hand, using the condition (A λ ), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equai_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaj_HTML.gif
Making use of Proposition 1 together with (20), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ31_HTML.gif
(31)
In order to derive the second estimate, we use the fact that, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq51_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq54_HTML.gif for i = 1,2, μ (1,2) and the condition (A λ ), then there exist c κ > 0 (only depend on κ) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equak_HTML.gif
Similarly, making use of Proposition 1 together with (19), we conclude that there exists https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (only depend on κ) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ32_HTML.gif
(32)

Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq55_HTML.gif for all λ (0, λ κ ) and ε (0, ε κ ). Then, (31) and (32) are enough to show that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq56_HTML.gif is a contraction from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq57_HTML.gif into itself and hence has a unique fixed point https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq58_HTML.gif in this set. This fixed point is solution of (20) in 2. We summarize this in the following:

Proposition 4 Given κ > 0, there exist ε κ > 0, λ κ > 0 and c κ > 0 (only depending on κ) such that for all ε (0, ε κ ), λ (0, λ κ ) satisfying (A), for all τ in some fixed compact subset of [τ -, τ+] (0, ∞) and for a given φ satisfying (24)-(30), then there exists a unique https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq59_HTML.gif solution of (29) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equal_HTML.gif
Solve (27) in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . In addition,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equam_HTML.gif

Observe that the function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq59_HTML.gif being obtained as a fixed point for contraction mappings, it depends continuously on the parameter τ.

4 The nonlinear exterior problem

Recall that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq60_HTML.gif denote the unique solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equan_HTML.gif
in Ω, with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq61_HTML.gif on ∂Ω. In addition, the following decomposition holds
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equao_HTML.gif
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq62_HTML.gif is a smooth function. Here, we give an estimate of the gradient of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq63_HTML.gif without proof (see [14], Lemma 2.1), there exists a constant c > 0, so that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equap_HTML.gif
Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq64_HTML.gif close enough to x := (x1, ..., x m ), https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq65_HTML.gif close to 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq66_HTML.gif satisfying (26). We define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ33_HTML.gif
(33)

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq67_HTML.gif is a cutoff function identically equal to 1 in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq68_HTML.gif and identically equal to 0 outside https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq69_HTML.gif .

We would like to find a solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ34_HTML.gif
(34)
in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq70_HTML.gif which is a perturbation of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq71_HTML.gif . Writing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq72_HTML.gif . This amounts to solve
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaq_HTML.gif

We need to define some auxiliary weighted spaces:

Definition 6 Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq73_HTML.gif , k , β (0, 1) and ν , we define the Hölder weighted space https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq74_HTML.gif as the set of functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq75_HTML.gif for which the following norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equar_HTML.gif

is finite

For all σ (0, r0/2) and all Y = (y1, ..., y m ) Ω m such that ||X - Y || ≤ r0/2, where X = (x1, ..., x m ), we denote by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equas_HTML.gif
the extension operator defined by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq76_HTML.gif in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq77_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equat_HTML.gif
for each i = 1, ..., m and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq78_HTML.gif in each Bσ/2(y i ), where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq79_HTML.gif is a cutoff function identically equal to 1 for t ≥ 1 and identically equal to 0 for t ≤ 1/2. It is easy to check that there exists a constant c = c(ν) > 0 only depending on ν such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ35_HTML.gif
(35)
We fix
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equau_HTML.gif
and denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq80_HTML.gif a right inverse of Δ provided by Proposition 3 with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq81_HTML.gif . Clearly, it is enough to find https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq82_HTML.gif solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ36_HTML.gif
(36)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equav_HTML.gif
We denote by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq83_HTML.gif the nonlinear operator which appears on the right hand side of Eq.(36). Given κ > 0 (whose value will be fixed later on), we assume that the points https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq84_HTML.gif , the functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq85_HTML.gif and the parameters https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq86_HTML.gif to satisfy
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ37_HTML.gif
(37)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ38_HTML.gif
(38)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ39_HTML.gif
(39)

Then, the following result holds

Lemma 6 Given κ > 0, there exist ε κ > 0, λ κ > 0, c κ > 0 and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (depending on κ) such that for all ε (0, ε κ ), λ (0, λ κ )
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaw_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equax_HTML.gif

provided https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq87_HTML.gif and satisfy https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq88_HTML.gif .

Proof: Recall that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq89_HTML.gif , we will estimate https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq90_HTML.gif in different subregions of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq91_HTML.gif .

* In https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq92_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq93_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq94_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ40_HTML.gif
(40)
so that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equay_HTML.gif
Hence, for ν (- 1, 0) and for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif small enough, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaz_HTML.gif
* In https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq96_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq94_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equba_HTML.gif
So, for ν (- 1, 0), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbb_HTML.gif
* In https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq97_HTML.gif , using the estimat (40), then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbc_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbd_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Eqube_HTML.gif
So,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ41_HTML.gif
(41)
Making use of Proposition 3 together with (34), we conclude that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ42_HTML.gif
(42)
For the proof of the second estimate, let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq99_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq100_HTML.gif for i = 1,2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbf_HTML.gif
Then for γ (0,1), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbg_HTML.gif
So, for https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif small enough and using the estimate (35), there exist https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq101_HTML.gif (depending on κ ), such that:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ43_HTML.gif
(43)

Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq55_HTML.gif for all λ (0, λ κ ) and ε (0, ε κ ). Then, (42) and (43) are enough to show that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq102_HTML.gif is a contraction from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq103_HTML.gif into itself and hence has a unique fixed point https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq104_HTML.gif in this set. This fixed point is solution of (35). We summarize this in the following

Proposition 5 Given κ > 0, there exists ε κ > 0 and λ κ > 0 (depending on κ) such that for all ε (0, ε κ ) and λ (0, λ κ ), for all set of parameter https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif satisfying (39) and function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq105_HTML.gif satisfying (26), there exists a unique https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq106_HTML.gif solution of (36) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbh_HTML.gif

As in the previous section, observe that the function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq107_HTML.gif being obtained as a fixed point for contraction mapping, depends smoothly on the parameters https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq108_HTML.gif and the points https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq109_HTML.gif .

5 The nonlinear Cauchy-data matching

Keeping the notations of the previous sections, we gather the results of Proposition 4 and 5. Assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq110_HTML.gif Ω m are given close to x := (x1, ..., x m ) and satisfy (37). Assume also that τ := (τ1, ..., τ m ) [τ - , τ +] m (0, ∞) m are given (the values of τ- and τ + will be fixed shortly). First, we consider some set of boundary data https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq111_HTML.gif satisfying (24). We set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbi_HTML.gif
According to the result of Proposition 4, we can find https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq112_HTML.gif a solution of
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ44_HTML.gif
(44)
in each https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq113_HTML.gif that can be decomposed as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbj_HTML.gif
where the function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq114_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ45_HTML.gif
(45)
Similarly, given some boundary data https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq115_HTML.gif satisfying (26), some parameters https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq116_HTML.gif satisfying (38), provide ε (0, ε κ ) and λ (0, λ κ ), we use the result of Proposition 5, to find a solution v ext of (43) which can be decomposed as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbk_HTML.gif
in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq117_HTML.gif where, the function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq118_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ46_HTML.gif
(46)
It remains to determine the parameters and the functions in such a way that the function which is equal to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq119_HTML.gif in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq120_HTML.gif and that is equal to vext in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq121_HTML.gif is a smooth function. This amounts to find the boundary data and the parameters so that, for each i = 1 ..., m
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ47_HTML.gif
(47)
on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif . Assuming we have already done so, this provides for each ε and λ small enough a function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq123_HTML.gif (which is obtained by patching together the functions https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq119_HTML.gif and the function vext) solution of -Δv - λ |v|2 = ρ2 (e v + e γv ) and elliptic regularity theory implies that this solution is in fact smooth. This will complete the proof of our result since, as ε and λ tend to 0, the sequence of solutions we have obtained satisfies the required properties, namely, away from the points x i the sequence vε,λconverges to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq124_HTML.gif . Before we proceed, the following remarks are due. First, it will be convenient to observe that the function https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq125_HTML.gif can be expanded as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ48_HTML.gif
(48)
near https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq126_HTML.gif . The function
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbl_HTML.gif
which appear in the expression of v ext can be expanded as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ49_HTML.gif
(49)
Near https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif . Here, we have defined
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbm_HTML.gif
Thus for x near https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq126_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ50_HTML.gif
(50)

where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq127_HTML.gif .

Next, in (47), all functions are defined on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif , but it will be convenient to solve the following equations
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ51_HTML.gif
(51)

on S1. Here, all functions are considered as functions of y S1 and we have simply used the change of variables https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq128_HTML.gif to parameterize https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif .

Since the boundary data, we have chosen satisfy (24) and (26), we can decompose
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbn_HTML.gif
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq129_HTML.gif are constant functions on S1, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq130_HTML.gif belong to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq131_HTML.gif and where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq132_HTML.gif are L2(S1) orthogonal to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif . Projecting the equations (51) over https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq133_HTML.gif will yield the system
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ52_HTML.gif
(52)
Let us comment briefly on how these equations are obtained. They simply come from (51) when expansions (48) and (49) are used, together with the expression of H i and H e given in Lemma 2 and Lemma 3, and also the estimates (45) and (46). The system (52) can be readily simplified into
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbo_HTML.gif
We are now in a position to define τ - and τ + since, according to the above, as ε and λ tend to 0 we expect that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq135_HTML.gif will converge to x i and that τ i will converge to https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq136_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbp_HTML.gif
and hence, it is enough to choose τ - and τ + in such a way that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbq_HTML.gif
We now consider the L2-projection of (51) over https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif . Given a smooth function f defined in Ω, we identify its gradient https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq137_HTML.gif with the element of https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbr_HTML.gif
With these notations in mind, we obtain the equations
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ53_HTML.gif
(53)
Finally, we consider the L2-projection onto L2(S1). This yields the system
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ54_HTML.gif
(54)
Thanks to the result of Lemma 4, this last system can be re-written as
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbs_HTML.gif
If we define the parameters t = (t i ) m by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbt_HTML.gif
then, the system we have to solve reads
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ55_HTML.gif
(55)

where as usual, the terms https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq138_HTML.gif depend nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and λ) time https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq139_HTML.gif , provide ε (0, ε κ ) and λ (0, λ κ ). Then, the nonlinear mapping which appears on the right-hand side of (55) is continuous and compact. In addition, reducing ε κ and λ κ if necessary, this nonlinear mapping sends the ball of radius https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq140_HTML.gif (for the natural product norm) into itself, provided κ is fixed large enough. Applying Schauder's fixed Theorem in the ball of radius https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq140_HTML.gif in the product space where the entries live yields the existence of a solution of Eq. (55) and this completes the proof of our Theorem 1. □

Declarations

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No RGP-VPP-087.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, King Saud University
(2)
Département de Mathématiques, Faculté des Sciences de Tunis Campus Universitaire

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