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.
Keywordssingular limits Green's function nonlinear Cauchy-data matching method
1 Introduction and statement of the results
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 Construction of the approximate solution
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 .
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:
2.2 Analysis of the Laplace operator in weighted spaces
With these notations, we have:
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.
3 The nonlinear interior problem
We denote by , the nonlinear operator appearing on the right-hand side of equation (29).
Then, we have the following result
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:
4 The nonlinear exterior problem
We need to define some auxiliary weighted spaces:
Then, the following result holds
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
5 The nonlinear Cauchy-data matching
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.
- 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
- 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
- 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
- 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
- Liouville J:Sur l'équation aux différences partielles . J Math 1853, 18: 17-72.Google Scholar
- Tarantello G: Multiple condensate solutions for the Chern-Simons-Higgs theory. J Math Phys 1996, 37: 3769-3796.View ArticleMathSciNetGoogle Scholar
- 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
- Nangasaki K, Suzuki T: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities. Asymptotic Anal 1990, 3: 173-188.MathSciNetGoogle Scholar
- 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
- Weston VH: On the asymptotique solution of a partial differential equation with exponential nonlinearity. SIAM J Math 1978, 9: 1030-1053.View ArticleMathSciNetGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- Wente HC: Counter example to a conjecture of H. Hopf. Pacific J Math 1986, 121: 193-243.View ArticleMathSciNetGoogle Scholar
- 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
- 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
- Clapp M, Munoz C, Musso M: Singular limits for the bi-Laplacian operator with exponential nonlinearity in ℝ4. Ann I H Poincaré AN 2008, 25: 1015-1041.View ArticleMathSciNetGoogle Scholar
- 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