Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation.

Let a > 0 , b > 0 and V (x) ≥ 0 be a coercive function in R 2 . We study the solutions with normalized L 2 -norm for the following Kirchhoff type equation - a + b ∫ R 2 | ∇ u | 2 d x Δ u + V (x) u = β | u | 2 u + λ u on a suitable weighted Sobolev space H = u ∈ H 1 (R 2) : ∫ R 2 V (x) u 2 d x < ∞. Our aim is to investigate the limit behaviors of the solutions with normalized L 2 -norm for this equation as (a , b) → (0 , 0) . Moreover, the uniqueness of the solution with normalized L 2 -norm for this equation is also discussed for a, b close to 0 [ABSTRACT FROM AUTHOR]