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 > 0 such that (0.1) has a ground state u λ 0 for any λ ∈ ( 0 , λ 0 ) , which blows up as λ → 0 + , but if V ( x ) = V ( | x | ) ∼ ( log ⁡ | x | ) α with α ∈ ( 0 , 1 ) at infinity, then there exists a sequence λ n → 0 + such that each u λ n 0 must be non-radially symmetric. However, if V ( x ) grows like | x | γ ( γ > 0) at infinity, we show that the functional I λ is of C 1 and (0.1) has a ground state u λ 1 for each λ < 0 which is radially symmetric. Also, the limit behavior of u λ 1 as λ → 0 − is discussed.