Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions - Qian, Xiaotao (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem \[ cases -(ε^2a+εb_R^3 | u|^2dx)Δ u+u=Q(x)|u|^p-2u, & xR^3, uH^1(R^3), u> 0, & xR^3, cases \] where ε> 0 is a small parameter, a,b> 0 are constants, 4< p< 6, Q is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of Q(x) affects the number of positive solutions.