Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R3. (February 2017) - Luo, Xiao; Wang, Qingfang
Abstract: In this paper, we study the multiplicity of solutions with a prescribed L 2 -norm for a class of nonlinear Kirchhoff type problems in R 3 − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u − λ u = | u | p − 2 u, where a, b > 0 are constants, λ ∈ R, p ∈ ( 14 3, 6 ) . To get such solutions we look for critical points of the energy functional I b ( u ) = a 2 ∫ R 3 | ∇ u | 2 + b 4 ( ∫ R 3 | ∇ u | 2 ) 2 − 1 p ∫ R 3 | u | p restricted on the following set S r ( c ) = { u ∈ H r 1 ( R 3 ) : ‖ u ‖ L 2 ( R 3 ) 2 = c },   c > 0 . For the value p ∈ ( 14 3, 6 ) considered, the functional I b is unbounded from below on S r ( c ) . By using a minimax procedure, we prove that for any c > 0, there are infinitely many critical points { u n b } n ∈ N + of I b restricted on S r ( c ) with the energy I b ( u n b ) → + ∞ ( n → + ∞ ) . Moreover, we regard b as a parameter and give a convergence property of u n b as b → 0 + .