The existence and concentration of positive solutions for a nonlinear Schr?dinger-Poisson system with critical growth

摘要

We consider the Schr?dinger-Poisson system: -ε~2?u + V(x)u + φ(x)u = f (u),-?φ = u~2 in R~3, where the nonlinear term f is of critical growth. In this paper, we construct a solution (u_ε, φ_ε) of the above elliptic system, which concentrates at an isolated component of positive locally minimum points of V as ε → 0 under certain conditions on f. In particular, the monotonicity of f (s)/s~3 and the so-called Ambrosetti-Rabinowitz condition are not required.