Multiple Solutions for Logarithmic Schrödinger–Poisson Systems with a Small Perturbation.

We consider the following logarithmic Schrödinger–Poisson system: - Δ u + ϕ u = u p - 2 u ln u 2 + λ f (x , u) , in Ω , - Δ ϕ = u 2 , in Ω , ϕ , u = 0 , on ∂ Ω , where Ω is a bounded domain in R 3 with smooth boundary ∂ Ω , p ∈ (4 , 6) , f(x, u) is continuous without any other condition. Using constrained variational method, Mountain Pass Theorem and iterative technique, we prove the existence of mountain pass solutions when λ > 0 small enough. Moreover, with f (x , 0) ≠ 0 in Ω , the above system possesses the another local minimum nontrivial solution. Finally, we prove that for any j ∈ N , there exists λ j > 0 , such that if 0 < λ < λ j , the above system possesses at least j distinct high energy solutions. [ABSTRACT FROM AUTHOR]