A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents

In the present paper, we discuss a Schrödinger–Kirchhoff equation involving the p(x)-Laplacian in the entire space RN. The primary focus of this article is on subcritical and critical nonlinearities. We deduce the existence of solutions by employing the mountain pass theorem in two distinct scenarios. Firstly, we discuss the equation when the potential function satisfies a weaker condition in the subcritical case. Secondly, we address the lack of compactness in the critical case without utilizing the concentration compactness principle.