Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term
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Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term Sami Baraket1*, Imed Abid2, Taieb Ouni2 and Nihed Trabelsi2 * Correspondence: [email protected] 1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Full list of author information is available at the end of the article
Abstract Given Ω bounded open regular set of ℝ2 and x1, x2, ..., xm Î Ω, we give a sufficient condition for the problem −div (eλu ∇u) = ρ 2 f (u)
to have a positive weak solution in Ω with u = 0 on ∂Ω, which is singular at each xi as the parameters r, l > 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 We consider the following problem
−div(a(u)∇u) u
= =
ρ 2 f (u) 0
in on
∂,
(1)
where ∇ is the gradient and Ω is an open smooth bounded subset of ℝ2. The function a is assumed to be positive and smooth. In the following, we take a(u) = elu and f (u) = elu(eu + egu), for l > 0 and g Î(0, 1), then problem (1) take the form −u − λ|∇u|2 = ρ 2 (eu + eγ u ) in ⊂ R2 (2) u = 0 on ∂. Using the following transformation w = (λρ 2 eu )λ ,
then the function w satisfies the following problem γ −1 λ+1 −w = w λ + w λ in ⊂ R2 λ w = (λρ 2 ) on ∂.
(3)
© 2011 Baraket et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Baraket et al. Boundary Value Problems 2011, 2011:10 http://www.boundaryvalueproblems.com/content/2011/1/10
Page 2 of 17
with ϱ = (lr2)1-l. So when l ® 0+, the exponent q = exponent
γ −1 λ
λ+1 λ
tends to infinity while the
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 ρ2 =
8 ε2 (1 + ε2 )2
.
(4)
Remark that r ~ ε as ε ® 0. We will suppose in the following (Aλ ) :
If
0 < ε < λ,
then
λ1+δ/2 ε−δ → 0
as
λ → 0, for any δ ∈ (0, 1).
In particular, if we take λ = O(ε2/3 ), then the condition (Al) is satisfied. Under the assumption (Al), we can treat equation (2) as a perturbation of the following: −u = ρ 2 (eu + eγ u )
in ⊂ R2
for g Î (0, 1). Our question is: Does there exist vε,l a sequence of solutions of (2) which converges to some singular function as the parameters ε and l tend to 0? In [3], Baraket et al. gave a positive answer to the above question for the following problem −u − λ|∇u|2 = ρ 2 eu in (5) u= 0 on ∂, 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 (xi)1≤i≤m as r and l a small param
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