Positive ground states for nonlinear Schrödinger–Kirchhoff equations with periodic potential or potential well in 𝐑^3

This work is devoted to the nonlinear Schrödinger–Kirchhoff-type equation - ( a+b ∫ _ℝ^3|∇ u | ^2 dx ) Δ u+V(x)u=f(x,u), in ℝ^3, where a>0 , b≥ 0 , the nonlinearity f(x,· ) is 3-superlinear and the potential V is either periodic or exhibits a finite potential well. By the mountain pass theorem, Lions’ concentration-compactness principle, and the energy comparison argument, we obtain the existence of positive ground state for this problem without proving the Palais–Smale compactness condition.