Existence and uniqueness results for p(x)$p(x)$‐Laplacian‐like Schrödinger equations with convection term in RN$\mathbb {R}^{N}$

Abstract In this paper, we discuss a class of Schrödinger equations involving ‐Laplacian‐like operators and a convection term. We first establish a novel compact embedding result for variable exponent Sobolev spaces in . Subsequently, by using Galerkin methods in combination with the fixed point theorem and pseudomonotone operators theory, we derive several existence results for various types of solutions, including finite‐dimensional approximate solutions, generalized solutions, and weak solutions. Finally, under appropriate assumptions on the nonlinearity, we demonstrate the uniqueness of the weak solution.