Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system(Pλ){−Δu+(1+μg(x))u+λϕ(x)u=|u|p−1u,x∈R3,−Δϕ=u2,lim|x|→+∞ϕ(x)=0, where λ, μ   are positive parameters, p∈(1,5)p∈(1,5), g(x)∈L∞(R3)g(x)∈L∞(R3) is nonnegative and g(x)≡0g(x)≡0 on a bounded domain in R3R3. In this case, μg(x)μg(x) represents a potential well that steepens as μ   getting large. If μ=0μ=0, (Pλ)(Pλ) was well studied in Ruiz (2006) [18]. If μ≠0μ≠0 and g(x)g(x) is not radially symmetric, it is unknown whether (Pλ)(Pλ) has a nontrivial solution for p∈(1,2)p∈(1,2). By priori estimates and approximation methods we prove that (Pλ)(Pλ) with p∈(1,2)p∈(1,2) has a ground state if μ large and λ   small. In the meantime, we prove also that (Pλ)(Pλ) with p∈[3,5)p∈[3,5) has a nontrivial solution for any λ>0λ>0 and μ   large. Moreover, some behaviors of the solutions of (Pλ)(Pλ) as λ→0λ→0, μ→+∞μ→+∞ and |x|→+∞|x|→+∞ are discussed.