On a class of Schrödinger equations in RNRN with indefinite linear part

In this paper, the existence of nontrivial solutions to the semilinear Schrödinger equation: −Δu+V(x)u=f(x,u),u∈H1(RN) has been studied, where V(x)V(x) is a continuous function such that the spectrum of the operator −Δ+V(x) has a negative part, f(x,u)f(x,u) is a continuous function with subcritical exponent growth, in particular, the limit f(x,u)→f∞(u)f(x,u)→f∞(u) exists as |x|→∞|x|→∞ and f(x,u)