Multiple Normalized Solutions for Choquard equations involving Kirchhoff type perturbation - Liu, Zeng (Topological Methods in Nonlinear Analysis, 2019)
In this paper we study the existence of critical points of the C^1 functional equation* E(u)=a2_| abla u|^2dx+b4(_| abla u|^2dx)^2-12p_(I_α*|u|^p)|u|^pdx equation* under the constraint equation* S_c={uH^1() _|u|^2dx=c^2}, equation* where a> 0, b> 0, N≥3, α(0,N), (N+α)/N< p< (N+α)/ (N-2) and I_α is the Riesz Potential. When p belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of E on S_c. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when c→ 0 and c→ +∞, respectively.