- Open Access
Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term
© Baraket et al; licensee Springer. 2011
- Received: 22 March 2011
- Accepted: 12 August 2011
- Published: 12 August 2011
Given Ω bounded open regular set of ℝ2 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.
- singular limits
- Green's function
- nonlinear Cauchy-data matching method
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?
with a regular bounded domain Ω of ℝ2. 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≤mas 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.
The study of this problem goes back to 1853 when Liouville derived a representation formula for all solutions of (7) which are defined in ℝ2, see . 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 . 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  which characterizes the possible limit of nontrivial branches of solutions of (7). His result has been generalized in  to (6) with , and finally by Ye in  to any exponentially dominated nonlinearity f(u). The existence of nontrivial branches of solutions with single singularity was first proved by Weston  and then a general result has been obtained by Baraket and Pacard . These results were also extended, applying to the Chern-Simons vortex theory in mind, by Esposito et al.  and Del Pino et al.  to handle equations of the form -Δu = ρ2V(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  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  and the problem with an additional singular source term given by Dirac masses is treated in  in the radial case. The results in [18, 19] are generalized to noncritical points of the reduced function, see .
which is well defined in (Ω) m for x i ≠ x j for i ≠ j. Our main result is the following
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  in the context of partial differential equations.
2.1 A linearized operator on ℝ2
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 ℝ2.
Lemma 1  Any bounded solution of defined in ℝ2 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 ℝ2.
As a consequence of the result of Lemma 1, we recall the surjectivity result of given in .
We set , we define
In . We will need the following:
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:
solves (16) in .
2.2 Analysis of the Laplace operator in weighted spaces
With these notations, we have:
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 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 denote by e1, e2 the coordinate functions on S1.
which decays at infinity.
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:
where H i (= H i (ψ; ·)) and H e = H e (ψ; ·), is an isomorphism.
We denote by , the nonlinear operator appearing on the right-hand side of equation (29).
Then, we have the following result
provided satisfying .
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 ℝ2. We summarize this in the following:
Observe that the function being obtained as a fixed point for contraction mappings, it depends continuously on the parameter τ.
where is a cutoff function identically equal to 1 in and identically equal to 0 outside .
We need to define some auxiliary weighted spaces:
Then, the following result holds
provided and satisfy .
Proof: Recall that , we will estimate in different subregions of .
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
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 .
on S1. Here, all functions are considered as functions of y ∈ S1 and we have simply used the change of variables to parameterize .
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. □
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.
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