Analysis of boundary bubbling solutions for an anisotropic Emden-Fowler equation
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چکیده
We consider the following anisotropic Emden-Fowler equation ∇(a(x)∇u) + εa(x)e = 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R is a smooth bounded domain and a is a positive smooth function. We study here the phenomenon of boundary bubbling solutions which do not exist for the isotropic case a ≡ constant. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that if x̄ ∈ ∂Ω is a strict local minimum point of a, there exists a family of solutions such that εa(x)edx tends to 8πa(x̄)δx̄ in D′(R2) as ε→ 0. This result will enable us to get a new family of solutions for the isotropic problem ∆u + εe = 0 in rotational torus of dimension N ≥ 3.
منابع مشابه
Bubbling Solutions for an Anisotropic Emden-Fowler Equation
We consider the following anisotropic Emden-Fowler equation ∇(a(x)∇u) + εa(x)e = 0 in Ω, u = 0 on ∂Ω where Ω ⊂ R is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quanti...
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تاریخ انتشار 2007