Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations inR3

Multiple and least energy sign-changing solutions for Schr¨odinger-Poisson equations in R3 with restraint

“…Luo and Wang in [31] proved that there are infinitely many normalized high energy solutions to Kirchhoff-type equations restrained on…”

Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations onRN

“…Very recently, some authors had studied the Kirchhoff type equation on the whole space R . Many solvability conditions with near zero and infinity for problem (2) have been considered, such as the superlinear case (see [19][20][21][22][23][24][25][26][27][28]); the asymptotically linear case (see [29,30]); the sublinear case (see [31][32][33]). Particularly, the following Kirchhoff type problem has been studied widely by some authors under various conditions on and :…”

Positive normalized solution to the Kirchhoff equation with general nonlinearities

“…If p = 2 + 8 N , the asymptotic behavior of critical points of I[u] on S c is also studied as c ↑ c. By scaling technique and energy estimate, Zeng and Zhang [55] improved the results of [53]. Precisely, they obtained the existence and uniqueness of minimizers of (1.13) with 0 < p < 2 + 8 N , and the existence and uniqueness of the mountain pass type critical points on the L 2 normalized manifold for 2 + 8 N < p < 2 * or p = 2 + 8 N and c > c. Using minimax procedure, Luo and Wang [38] obtained the multiplicity of critical points of I(u) on the L 2 normalized manifold for 2 + 8 N < p < 2 * and any c > 0. Recently, He, Lv, the third author and the fourth author [15] consider the general nonlinearity g of mass super critical case, they study the existence of ground state normalized solutions for any given c > 0 via Pohozaev manifold constraint method and also study the asymptotic behavior of these solutions as c → 0 + as well as c → +∞, via a sequence of blow up arguments.…”