Nonradial symmetry and blow-up of ground states for a Schrödinger–Poisson system with logarithmic term

Abstract We consider the following Schrödinger–Poisson system { − Δ u + V ( x ) u + λ ϕ ( x ) u = u log u 2 , x ∈ ℝ 3 , (0.1) − Δ ϕ = u 2 , lim | x | → + ∞ ϕ ( x ) = 0 , where λ ∈ R is a parameter, V ∈ C ( R 3 , R + ) is a coercive potential. We prove that, if V ( x ) ∼ ( log ⁡ | x | ) 1 2 at infinity, then the energy functional I λ associated with (0.1) fails to be C 1 , and there is λ 0