In this paper, we consider the following nonlinear Schrödinger equation with Sobolev critical growth: Formula: see text where Formula: see text and Formula: see text is the Sobolev critical exponent. The potential function Formula: see text changes sign but is not periodic, and Formula: see text satisfies the assumptions of superlinear and Sobolev subcritical growth at infinity. Under the indefinite framework, where the Schrödinger operator Formula: see text has negative eigenvalues, we establish the existence of ground state solutions for the above problem by employing an appropriate variational approach and utilizing information about the autonomous problem at infinity. Moreover, we show that any ground state solution is continuous and decays exponentially to zero as Formula: see text.
Long et al. (Sat,) studied this question.