Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrodinger-Poisson system in R3

We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrodinger-Poisson system {-Delta u + V(x)u + lambda phi(x)u = f(u), x is an element of R-3, -Delta phi = u(2), where V(x) is a smooth function and lambda is a positive parameter. Because the so-called nonlocal term lambda phi(u)(x)u is involving in the equation, the variational functional of the equation has totally different properties from the case of lambda - 0 . Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution u(lambda). Moreover, we show that any sign-changing solution of the problem has an energy exceeding twice the least energy, and for any sequence {lambda(n)} -> 0(+) (n -> infinity), there is a subsequence {lambda(nk)}, such that u(lambda nk) converges in H-1 (R-3) to u(0) as k -> infinity, where u(0) is a sign-changing solution of the following equation -Delta u + V(x)u = f(u), x is an element of R-3.