An eigenvalue problem for nonlinear Schrödinger-Poisson system with
steep potential well

In this paper, we study an eigenvalue problem for Schrödinger-Poisson
system with indefinite nonlinearity and potential well as follows:
-Δu+μV(x)u+K(x)Φu =
λf(x)u+g(x)|u|^{p-2}u in
R^{3}, -ΔΦ=K(x)u^2 in R^{3}, where
4≤p<6, the parameters μ, λ>0, V∈
C(R^{3}) is a potential well, and the functions f ∈
L^{3/2}(R^{3}) and g ∈
L^{∞}(R^{3}) are allowed to be
sign-changing. It is well known that such a system with the potential
being positive constant has two positive solutions when
lim_{|x|→∞}g(x)=g_{∞}<0,
K=0 in the set {x ∈ R^{3} : g(x)=0} and
λ>λ_{1}(f) with near λ_{1}(f),
where λ_{1}(f) is the first eigenvalue of -Δ+ id in
H^{1}(R^{3}) (see e.g. Huang et al., J.
Differential Equations 255, 2463 (2013)). The main purpose is to obtain
the existence and multiplicity of positive solutions without the above
assumptions for g and K. The results are obtained via variational method
and steep potential. Furthermore, we also consider the concentration of
solutions as μ→∞.

2021