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

- Sami Baraket
^{1}Email author, - Imed Abid
^{2}, - Taieb Ouni
^{2}and - Nihed Trabelsi
^{2}

**2011**:10

**DOI: **10.1186/1687-2770-2011-10

© Baraket et al; licensee Springer. 2011

**Received: **22 March 2011

**Accepted: **12 August 2011

**Published: **12 August 2011

## 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 Cauchy-data matching method## 1 Introduction and statement of the results

^{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

*w*satisfies the following problem

with ϱ = (*λρ*^{2})^{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].

*ε*the smallest positive parameter satisfying

*ρ*~

*ε*as

*ε*→ 0. We will suppose in the following

*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?

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.

*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 Chern-Simons vortex theory in mind, by Esposito et al. [12] and Del Pino et al. [13] to handle equations of the form -Δ*u* = *ρ*^{2}*V*(*x*)*e*^{
u
} where *V* is a nonconstant positive potential. See also [14–16] 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].

*G*(

*x*,

*x*') defined on Ω × Ω, to be solution of

*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 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

^{2}which will play a central role in our analysis. Given

*ε*> 0, we define

^{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*·) + 2log

*τ*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}

*ε*,

*τ*,

*λ*> 0, we set

*δ*∈ (0, 1). We define the linear second order elliptic operator

*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

*r*= |

*x*|. Clearly and this reflects the fact that (11) is invariant under the group of dilations

*τ*→

*u*(

*τ*·) + 2 log

*τ*. 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*.

As a consequence of the result of Lemma 1, we recall the surjectivity result of given in [11].

*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*.

*u*of

*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*

*σ*≥ 1, we denote by the extension operator defined by

*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

*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

*κ*> 0, there exist

*c*

_{ κ }

*>*0 (only depend on

*κ*), such that for

*δ*∈ (0,1) and |

*x|*=

*r*, we have

*h*

_{1},

*h*

_{2}such that in , then for

*δ*∈ (0, 1 -

*r*] we have

*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

*x*

_{1}, ...,

*x*

_{ m }∈ Ω, we define

**x**:= (

*x*

_{1}, ...,

*x*

_{ m })

*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

*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

*H*

^{ i }(=

*H*

^{ i }(

*φ*; ·)) to be the solution of

We denote by *e*_{1}, *e*_{2} the coordinate functions on *S*^{1}.

*then there exists c*> 0

*such that*

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*.

*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:

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

## 3 The nonlinear interior problem

In .

*w*=

**v**+

*v*and using the fact that

*H*

^{ i }is harmonic, this amounts to solve

*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 right-hand side of equation (29).

*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

*δ*∈ (0, 1 -

*γ*]. Then

*A*

_{ λ }), we have

*i*= 1,2,

*μ*∈ (1,2) and the condition (

*A*

_{ λ }), then there exist

*c*

_{ κ }

*>*0 (only depend on

*κ*) such that

*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

*c >*0, so that

**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 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*

*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

*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

*κ*> 0 (whose value will be fixed later on), we assume that the points , the functions and the parameters to satisfy

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
.

*ν*∈ (- 1, 0) and for small enough, we get

*ν*∈ (- 1, 0), we have

*i*= 1,2, we have

*γ*∈ (0,1), we get

*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 Cauchy-data matching

^{ 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

*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

*v*

_{ext}in is a smooth function. This amounts to find the boundary data and the parameters so that, for each

*i*= 1 ...,

*m*

*ε*and

*λ*small enough a function (which is obtained by patching together the functions and the function

*v*

_{ext}) solution of -Δ

*v*-

*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

*v*

_{ ext }can be expanded as

*x*near , we have

where .

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
.

*S*

^{1}, belong to and where are

*L*

^{2}(

*S*

^{1}) orthogonal to and . Projecting the equations (51) over will yield the system

*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

*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

*τ*

_{ - }and

*τ*

^{+}in such a way that

*L*

^{2}-projection of (51) over . Given a smooth function

*f*defined in Ω, we identify its gradient with the element of

*L*

^{2}-projection onto

*L*

^{2}(

*S*

^{1})

^{⊥}. This yields the system

**t**= (

*t*

_{ i }) ∈ ℝ

^{ m }by

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

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