Multiplicity of Solutions for an Integro-Differential Equation with Trudinger-Moser Subcritical or Critical Nonlinearities

Abstract

This paper study a general nonlocal problem characterized by the equation:

\[
-\mathcal{L}_K u + V(x)u = P(x)h(u)
\]

in $\mathbb{R}$.

Here, $\mathcal{L}_K$ represents a nonlocal integro-differential operator, and $h$ is a nonlinear term displaying subcritical or critical growth of Trudinger-Moser type. We initially prove the existence of a ground state solution by employing variational methods and innovative analytical approaches. Furthermore, through the application of constrained variational methods, Brouwer degree theory, and a quantitative deformation lemma, we establish the existence of a sign-changing solution with minimal energy, surpassing the energy of the ground state solution.