Abstract
Given Ω bounded open regular set of ℝ^{2 }and x_{1}, x_{2}, ..., x_{m }∈ Ω, we give a sufficient condition for the problem
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 Cauchydata matching method1 Introduction and statement of the results
We consider the following problem
where ∇ is the gradient and Ω is an open smooth bounded subset of ℝ^{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
Using the following transformation
then the function w satisfies the following problem
with ϱ = (λρ^{2})^{1λ}. So when λ → 0^{+}, the exponent tends to infinity while the exponent 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
Remark that ρ ~ ε as ε → 0. We will suppose in the following
In particular, if we take , then the condition (A_{λ}) is satisfied. Under the assumption (A_{λ}), we can treat equation (2) as a perturbation of the following:
for γ ∈ (0, 1).
Our question is: Does there exist v_{ε,λ }a 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
with a regular bounded domain Ω of ℝ^{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≤i≤m }as 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
for γ ∈ (0, 1) as ρ tends to 0. When λ = 0 and γ = 0, problem (2) reduce to
The study of this problem goes back to 1853 when Liouville derived a representation formula for all solutions of (7) which are defined in ℝ^{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 , 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 ChernSimons vortex theory in mind, by Esposito et al. [12] and Del Pino et al. [13] to handle equations of the form Δu = ρ^{2}V(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 4dimensional semilinear elliptic problem with biLaplacian 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
and let H(x, x') = G(x, x') + 4log x  x', its regular part. Let m ∈ ℕ, we set
which is well defined in (Ω)^{m }for x_{i }≠ x_{j }for i ≠ j. Our main result is the following
Theorem 1 Given β ∈ (0, 1). Let Ω an open smooth bounded set of ℝ^{2}, λ > 0 satisfying the condition (A_{λ}), γ ∈ (0, 1) and S = {x_{1}, ... x_{m}} ⊂ Ω be a nonempty set. Assume that, the point (x_{1}, ..., x_{m}) is a nondegenerate critical point of the function
then there exist ε_{0 }> 0, λ_{0 }> 0 and a family of solutions of (2), such that
One of the purpose of the present paper is to present a rather efficient method: nonlinear Cauchydata 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
in ℝ^{2 }which will play a central role in our analysis. Given ε > 0, we define
which is clearly a solution of
in ℝ^{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
2.1 A linearized operator on ℝ^{2}
For all ε, τ, λ > 0, we set
for δ ∈ (0, 1). We define the linear second order elliptic operator
which corresponds to the linearization of (11) about the solution u_{1 }(= u_{ε = τ = 1}) given by (10) which has been defined in the previous section. We are interested in the classification of bounded solutions of in ℝ^{2}. Some solutions are easy to find. For example, we can define
where r = x. Clearly and this reflects the fact that (11) is invariant under the group of dilations τ → u(τ ·) + 2 logτ. We also define, for i = 1, 2
which are also solutions of . Since, these solutions correspond to the invariance of the equation under the group of translations a → u(· + a). We recall the following result which classifies all bounded solutions of which are defined in ℝ^{2}.
Lemma 1 [11]Any bounded solution of defined in ℝ^{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 ℝ^{2}.
Definition 1 Given k ∈ ℕ, β ∈ (0, 1) and μ ∈ ℝ, we introduce the Hölder weighted spaces as the space of functions for which the following norm
is finite.
We define also
As a consequence of the result of Lemma 1, we recall the surjectivity result of given in [11].
Proposition 1 [11]
(i) Assume that μ > 1 and μ ∉ ℕ, then
is surjective.
(ii) Assume that δ > 0 and δ ∉ ℕ then
is surjective.
We set , we define
Definition 2 Given k ∈ ℕ, β ∈ (0, 1) and μ ∈ ℝ, we introduce the Hölder weighted spaces as the space of functions in for which the following norm
is finite.
Then, we define the subspace of radial functions in by
We would like to find a solution u of
in . By using the transformation, then Eq. (15) is equivalent to
in . We look for a solution of (16) of the form v(x) = u_{1}(x) + h(x), this amounts to solve
In . We will need the following:
Definition 3 Given , k ∈ ∞, β ∈ (0, 1) and μ ∈ ℝ, the weighted space is defined to be the space of functions endowed with the norm
For all σ ≥ 1, we denote by the extension operator defined by
where t α χ(t) is a smooth nonnegative 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
We fix δ ∈ (0, 1) and denote by to be a right inverse of provided by Proposition 1. To find a solution of (17), it is enough to find a fixed point h, in a small ball of , solution of
We have
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
Making use of Proposition 1 together with (19), we conclude that
Now, let h_{1}, h_{2 }such that in , then for δ ∈ (0, 1  r] we have
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 (only depend on κ) such that
Reducing λ_{κ }> 0 and ε_{κ }> 0 if necessary, we can assume that, for all λ ∈ (0, λ_{κ}) and ε ∈ (0, ε_{κ}). Then, (21) and (22) are enough to show that h ↦ ℵ is a contraction from into itself and hence has a unique fixed point h in this set. This fixed point is solution of (20) in . We summarize this in the:
Proposition 2 Given δ ∈ (0, 1  γ] and κ > 1, then there exist (independent of ε and λ) and a unique with such that
solves (16) in .
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 x_{1}, ..., x_{m }∈ Ω, we define x := (x_{1}, ..., x_{m})
and we choose r_{0 }> 0 so that the balls of center x_{i }and radius r_{0 }are mutually disjoint and included in Ω. For all r ∈ (0, r_{0}), we define
With these notations, we have:
Definition 4 Given k ∈ ℝ, β ∈ (0,1) and ν ∈ ℝ, we introduce the Hölder weighted space as the space of functions for with the following norm
is finite.
When k ≥ 2, we denote by be the subspace of functions satisfying w = 0 on ∂Ω. We recall the
Proposition 3 [21]Assume that ν < 0 and ν ∉ ℤ, then
is surjective. Denote by a right inverse of .
Remark 1 Observe that, when ν < 0, ν ∉ ℤ, the right inverse even though is not unique and can be chosen to depend smoothly on the points x_{1}, ..., x_{m}, at least locally. Once a right inverse is fixed for some choice of the points x_{1}, ..., x_{m}, a right inverse which depends smoothly on some points close to x_{1}, ..., 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 and define H^{i }(=H^{i }(φ; ·)) to be the solution of
We denote by e_{1}, e_{2 }the coordinate functions on S^{1}.
Lemma 2 [21]If we assume that
then there exists c > 0 such that
Given , we define to be the solution of
which decays at infinity.
Definition 5 Given k ∈ ℕ, β ∈ (0,1) and ν ∈ ℝ, we define the space as the space of functions for which the following norm
is finite.
Lemma 3 [21]If we assume that
Then there exists c > 0 such that
If F ⊂ L^{2}(S^{1}) is a space of functions defined on S^{1}, we define the space F_{⊥ }to be the subspace of functions F of which are L^{2}(S^{1}) orthogonal to the functions 1, e_{1},e_{2}. We will need the:
Lemma 4 [21]The mapping
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
In .
Given satisfying (24), we define
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
We fix μ ∈ (1,2) and denote by to be a right inverse of provided by Proposition 1. To find a solution of (28), it is sufficient to find solution of
We denote by , the nonlinear operator appearing on the righthand side of equation (29).
Given κ > 0 (whose value will be fixed later on), we further assume that the functions φ satisfy
Then, we have the following result
Lemma 5 Given κ > 0. There exist ε_{κ }> 0, λ_{κ }> 0, c_{κ }> 0 and (only depend on κ) such that for all λ ∈ (0, λ_{κ}) and ε ∈ (0, ε_{κ})
and
provided satisfying .
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
Since for each , we have
where δ ∈ (0, 1  γ]. Then
On the other hand, using the condition (A_{λ}), we have
and
Making use of Proposition 1 together with (20), we get
In order to derive the second estimate, we use the fact that, for satisfying for i = 1,2, μ ∈ (1,2) and the condition (A_{λ}), then there exist c_{κ }> 0 (only depend on κ) such that
Similarly, making use of Proposition 1 together with (19), we conclude that there exists (only depend on κ) such that
□
Reducing λ_{κ }> 0 and ε_{κ }> 0 if necessary, we can assume that, for all λ ∈ (0, λ_{κ}) and ε ∈ (0, ε_{κ}). Then, (31) and (32) are enough to show that is a contraction from into itself and hence has a unique fixed point in this set. This fixed point is solution of (20) in ℝ^{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 solution of (29) such that
Solve (27) in . In addition,
Observe that the function being obtained as a fixed point for contraction mappings, it depends continuously on the parameter τ.
4 The nonlinear exterior problem
Recall that denote the unique solution of
in Ω, with on ∂Ω. In addition, the following decomposition holds
where is a smooth function. Here, we give an estimate of the gradient of without proof (see [14], Lemma 2.1), there exists a constant c > 0, so that
Let close enough to x := (x_{1}, ..., x_{m}), close to 0 and satisfying (26). We define
where is a cutoff function identically equal to 1 in and identically equal to 0 outside .
We would like to find a solution of
in which is a perturbation of . Writing . This amounts to solve
We need to define some auxiliary weighted spaces:
Definition 6 Let , k ∈ ℝ, β ∈ (0, 1) and ν ∈ ℝ, we define the Hölder weighted space as the set of functions for which the following norm
is finite
For all σ ∈ (0, r_{0}/2) and all Y = (y_{1}, ..., y_{m}) ∈ Ω^{m }such that X  Y  ≤ r_{0}/2, where X = (x_{1}, ..., x_{m}), we denote by
the extension operator defined by in
for each i = 1, ..., m and in each B_{σ/2}(y_{i}), where 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
We fix
and denote by a right inverse of Δ provided by Proposition 3 with . Clearly, it is enough to find solution of
where
We denote by 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 , the functions and the parameters to satisfy
and
Then, the following result holds
Lemma 6 Given κ > 0, there exist ε_{κ }> 0, λ_{κ }> 0, c_{κ }> 0 and (depending on κ) such that for all ε ∈ (0, ε_{κ }), λ ∈ (0, λ_{κ})
and
provided and satisfy .
Proof: Recall that , we will estimate in different subregions of .
* In , we have , and
so that
Hence, for ν ∈ ( 1, 0) and for small enough, we get
* In , we have and . Thus
So, for ν ∈ ( 1, 0), we have
* In , using the estimat (40), then we have
where
Then
So,
Making use of Proposition 3 together with (34), we conclude that
For the proof of the second estimate, let and satisfying for i = 1,2, we have
Then for γ ∈ (0,1), we get
So, for small enough and using the estimate (35), there exist (depending on κ ), such that:
□
Reducing λ_{κ }> 0 and ε _{κ }> 0 if necessary, we can assume that, for all λ ∈ (0, λ_{κ}) and ε ∈ (0, ε_{κ}). Then, (42) and (43) are enough to show that is a contraction from into itself and hence has a unique fixed point 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 satisfying (39) and function satisfying (26), there exists a unique solution of (36) such that
As in the previous section, observe that the function being obtained as a fixed point for contraction mapping, depends smoothly on the parameters and the points .
5 The nonlinear Cauchydata matching
Keeping the notations of the previous sections, we gather the results of Proposition 4 and 5. Assume that ∈ Ω^{m }are given close to x := (x_{1}, ..., x_{m}) and satisfy (37). Assume also that τ := (τ_{1}, ..., 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 satisfying (24). We set
According to the result of Proposition 4, we can find a solution of
in each that can be decomposed as
where the function satisfies
Similarly, given some boundary data satisfying (26), some parameters 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
in where, the function satisfies
It remains to determine the parameters and the functions in such a way that the function which is equal to in and that is equal to v_{ext }in is a smooth function. This amounts to find the boundary data and the parameters so that, for each i = 1 ..., m
on . Assuming we have already done so, this provides for each ε and λ small enough a function (which is obtained by patching together the functions and the function v_{ext}) solution of Δv  λ ∇v^{2 }= ρ^{2 }(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 v_{ε,λ }converges to . Before we proceed, the following remarks are due. First, it will be convenient to observe that the function can be expanded as
near . The function
which appear in the expression of v_{ext }can be expanded as
Near . Here, we have defined
Thus for x near , we have
where .
Next, in (47), all functions are defined on , but it will be convenient to solve the following equations
on S^{1}. Here, all functions are considered as functions of y ∈ S^{1 }and we have simply used the change of variables to parameterize .
Since the boundary data, we have chosen satisfy (24) and (26), we can decompose
where are constant functions on S^{1}, belong to and where are L^{2}(S^{1}) orthogonal to and . Projecting the equations (51) over will yield the system
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
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 will converge to x_{i }and that τ_{i }will converge to satisfying
and hence, it is enough to choose τ _{ }and τ ^{+ }in such a way that
We now consider the L^{2}projection of (51) over . Given a smooth function f defined in Ω, we identify its gradient with the element of
With these notations in mind, we obtain the equations
Finally, we consider the L^{2}projection onto L^{2}(S^{1})^{⊥}. This yields the system
Thanks to the result of Lemma 4, this last system can be rewritten as
If we define the parameters t = (t_{i}) ∈ ℝ^{m }by
then, the system we have to solve reads
where as usual, the terms 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 , provide ε ∈ (0, ε_{κ}) and λ ∈ (0, λ_{κ}). Then, the nonlinear mapping which appears on the righthand 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 (for the natural product norm) into itself, provided κ is fixed large enough. Applying Schauder's fixed Theorem in the ball of radius 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. □
Competing interests
The authors declare that they have no competing interests.
Authors' contribution
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
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 RGPVPP087.
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