PROPERTIES OF THE MINIMIZERS FOR A CONSTRAINED MINIMIZATION PROBLEM ARISING IN KIRCHHOFF EQUATION

摘要

Let a ＞ 0, b ＞ 0 and V(ⅹ) ≥ 0 be a coercive function in R~2. We study the following constrained minimization problem on a suitable weighted Sobolev space H: e_a(b) := inf{E_a~b(u): u∈ H and ∫_(R~2) |u|~2dx = 1}, where E_a~b(u) is a Kirchhoff type energy functional defined on H by E_a~b(u) =1/2 ∫_(R~2)[|▽u|~2 + V(ⅹ)u~2}dx + b/4 ( ∫_(R~2)|▽u|~2dx)~2-a/4∫_(R~2)|u|~4dx. It is known that, for some a* ＞ 0, e_a(b) has no minimizer if b = 0 and a ≥ a*, but e_a(b) has always a minimizer for any a ≥ 0 if b ＞ 0. The aim of this paper is to investigate the limit behaviors of the minimizers of e_a(b) as b → 0~+. Moreover, the uniqueness of the minimizers of e_a (b) is also discussed for b close to 0.