Open Access

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

Boundary Value Problems20112011:10

https://doi.org/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

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 limitsGreen's functionnonlinear Cauchy-data matching method

1 Introduction and statement of the results

We consider the following problem
(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
(2)
Using the following transformation
then the function w satisfies the following problem
(3)

with ϱ = (λρ2)1-λ. 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
(4)
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 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
(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
(6)
for γ (0, 1) as ρ tends to 0. When λ = 0 and γ = 0, problem (2) reduce to
(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 , 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
and let H(x, x') = G(x, x') + 4log |x - x'|, its regular part. Let m , we set
(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
then there exist ε0 > 0, λ0 > 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 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
(9)
in 2 which will play a central role in our analysis. Given ε > 0, we define
(10)
which is clearly a solution of
(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
(12)

2.1 A linearized operator on 2

For all ε, τ, λ > 0, we set
(13)
for δ (0, 1). We define the linear second order elliptic operator
(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 in 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 τu(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 au(· + a). We recall the following result which classifies all bounded solutions of which are defined in 2.

Lemma 1 [11] Any bounded solution of 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 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]
  1. (i)
    Assume that μ > 1 and μ , then
     
is surjective.
  1. (ii)
    Assume that δ > 0 and δ then
     

is surjective.

We set , we define

Definition 2 Given k , β (0, 1) and μ , 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
(15)
in . By using the transformation, then Eq. (15) is equivalent to
(16)
in . We look for a solution of (16) of the form v(x) = u1(x) + h(x), this amounts to solve
(17)

In . We will need the following:

Definition 3 Given , k, β (0, 1) and μ , 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
(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
(19)
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
(20)
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
(21)
Now, let h1, h2 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
(22)

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 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 x1, ..., x m Ω, we define x := (x1, ..., x m )
and we choose r0 > 0 so that the balls of center x i and radius r0 are mutually disjoint and included in Ω. For all r (0, r0), we define

With these notations, we have:

Definition 4 Given k , β (0,1) and ν , 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 , then

is surjective. Denote by a right inverse of .

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 and define H i (=H i (φ; ·)) to be the solution of
(23)

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

Lemma 2 [21] If we assume that
(24)
then there exists c > 0 such that
Given , we define to be the solution of
(25)

which decays at infinity.

Definition 5 Given k , β (0,1) and ν , we define the space as the space of functions for which the following norm

is finite.

Lemma 3 [21] If we assume that
(26)
Then there exists c > 0 such that

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

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
(27)

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
(28)
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
(29)

We denote by , 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
(30)

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
(31)
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
(32)

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 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 := (x1, ..., x m ), close to 0 and satisfying (26). We define
(33)

where is a cutoff function identically equal to 1 in and identically equal to 0 outside .

We would like to find a solution of
(34)
in which is a perturbation of . Writing . This amounts to solve

We need to define some auxiliary weighted spaces:

Definition 6 Let , k , β (0, 1) and ν , we define the Hölder weighted space as the set of functions for which the following norm

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
the extension operator defined by in
for each i = 1, ..., m and in each Bσ/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
(35)
We fix
and denote by a right inverse of Δ provided by Proposition 3 with . Clearly, it is enough to find solution of
(36)
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
(37)
(38)
and
(39)

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
(40)
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,
(41)
Making use of Proposition 3 together with (34), we conclude that
(42)
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:
(43)

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 Cauchy-data 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 := (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 satisfying (24). We set
According to the result of Proposition 4, we can find a solution of
(44)
in each that can be decomposed as
where the function satisfies
(45)
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
(46)
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 vext in is a smooth function. This amounts to find the boundary data and the parameters so that, for each i = 1 ..., m
(47)
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 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 . Before we proceed, the following remarks are due. First, it will be convenient to observe that the function can be expanded as
(48)
near . The function
which appear in the expression of v ext can be expanded as
(49)
Near . Here, we have defined
Thus for x near , we have
(50)

where .

Next, in (47), all functions are defined on , but it will be convenient to solve the following equations
(51)

on S1. Here, all functions are considered as functions of y S1 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 S1, belong to and where are L2(S1) orthogonal to and . Projecting the equations (51) over will yield the system
(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
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 L2-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
(53)
Finally, we consider the L2-projection onto L2(S1). This yields the system
(54)
Thanks to the result of Lemma 4, this last system can be re-written as
If we define the parameters t = (t i ) m by
then, the system we have to solve reads
(55)

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 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 (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. □

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

References

  1. Ren X, Wei J: On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans Am Math Soc 1994, 343: 749-763.View ArticleMathSciNetGoogle Scholar
  2. Esposito P, Musso M, Pistoia A: Concentrating solutions for a planar problem involving nonlinearities with large exponent. J Diff Eqns 2006, 227: 29-68.View ArticleMathSciNetGoogle Scholar
  3. Baraket S, Ben Omrane I, Ouni T: Singular limits solutions for 2-dimensional elliptic problem involving exponential nonlinearities with non linear gradient term. Nonlinear Differ Equ Appl 2011, 18: 59-78.View ArticleMathSciNetGoogle Scholar
  4. Baraket S, Ye D: Singular limit solutions for two-dimensional elliptic problems with exponentionally dominated nonlinearity. Chin Ann Math Ser B 2001, 22: 287-296.View ArticleMathSciNetGoogle Scholar
  5. Liouville J:Sur l'équation aux différences partielles . J Math 1853, 18: 17-72.Google Scholar
  6. Tarantello G: Multiple condensate solutions for the Chern-Simons-Higgs theory. J Math Phys 1996, 37: 3769-3796.View ArticleMathSciNetGoogle Scholar
  7. Suzuki T: Two dimensional Emden-Fowler Equation with Exponential Nonlinearity. Nonlinear Diffusion Equations and Their Equilibrium States Birkäuser 1992, 3: 493-512.View ArticleGoogle Scholar
  8. Nangasaki K, Suzuki T: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Anal 1990, 3: 173-188.MathSciNetGoogle Scholar
  9. Ye D: Une remarque sur le comportement asymptotique des solutions de - Δ u = λ f ( u ). C R Acad Sci Paris I 1997, 325: 1279-1282.View ArticleGoogle Scholar
  10. Weston VH: On the asymptotique solution of a partial differential equation with exponential nonlinearity. SIAM J Math 1978, 9: 1030-1053.View ArticleMathSciNetGoogle Scholar
  11. Baraket S, Pacard F: Construction of singular limits for a semilinear elliptic equation in dimension 2. Calc Var Partial Differ Equ 1998, 6: 1-38.View ArticleMathSciNetGoogle Scholar
  12. Esposito P, Grossi M, Pistoia A: On the existence of Blowing-up solutions for a mean field equation. Ann I H Poincaré -AN 2005, 22: 227-257.View ArticleMathSciNetGoogle Scholar
  13. Del Pino M, Kowalczyk M, Musso M: Singular limits in Liouville-type equations. Calc Var Partial Differ Equ 2005, 24: 47-87.View ArticleMathSciNetGoogle Scholar
  14. Wei J, Ye D, Zhou F: Bubbling solutions for an anisotropic Emden-Fowler equation. Calc Var Partial Differ Equ 2007, 28: 217-247.View ArticleMathSciNetGoogle Scholar
  15. Wei J, Ye D, Zhou F: Analysis of boundary bubbling solutions for an anisotropic Emden-Fowler equation. Ann I H Poincaré AN 2008, 25: 425-447.View ArticleMathSciNetGoogle Scholar
  16. Ye D, Zhou F: A generalized two dimensional Emden-Fowler equation with exponential nonlinearity. Calc Var Partial Differ Equ 2001, 13: 141-158.View ArticleMathSciNetGoogle Scholar
  17. Wente HC: Counter example to a conjecture of H. Hopf. Pacific J Math 1986, 121: 193-243.View ArticleMathSciNetGoogle Scholar
  18. Baraket S, Dammak M, Ouni T, Pacard F: Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity. Ann I H Poincaré AN 2007, 24: 875-895.View ArticleMathSciNetGoogle Scholar
  19. Dammak M, Ouni T: Singular limits for 4-dimensional semilinear elliptic problem with exponential nonlinearity adding a singular source term given by Dirac masses. Differ Int Equ 2008, 11-12: 1019-1036.MathSciNetGoogle Scholar
  20. Clapp M, Munoz C, Musso M: Singular limits for the bi-Laplacian operator with exponential nonlinearity in 4. Ann I H Poincaré AN 2008, 25: 1015-1041.View ArticleMathSciNetGoogle Scholar
  21. Baraket S, Ben Omrane I, Ouni T, Trabelsi N: Singular limits solutions for 2-dimensional elliptic problem with exponentially dominated nonlinearity and singular data. Communications in Contemporary Mathematics 2 2011, 13(4):129.MathSciNetGoogle Scholar

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© Baraket et al; licensee Springer. 2011

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