Ground state solutions for a non-periodic indefinite variational problems with Sobolev critical growth

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].