This study focused on establishing the existence and multiplicity of solutions for gauged nonlinear Schrödinger equations set on the plane with sign-changing potentials. Our findings contribute to the extension of recent advancements in this area of research. Initially, we examined scenarios where the potential function $V$ is lower-bounded and the function space has a compact embedding into Lebesgue spaces. Subsequently, we addressed more complex cases characterized by a sign-changing potential $V$ and a function space that fails to compactly embed into Lebesgue spaces. The proofs of our results are based on the Trudinger-Moser inequality, the application of variational methods, and the utilization of Morse theory.

In this paper, We studied a class of fractional $p$-Laplacian equations with nonsmooth potential. By nonsmooth analysis, nonsmooth critical point theory, and truncation techniques, we established two multiplicity theorems: One guaranteeing the existence of at least four solutions, and the other ensuring the existence of at least two solutions. In our approach, we addressed the challenges posed by the nonsmoothness of the potential, and the results extended the nonlocal problems with discontinuous nonlinearities.