Existence and multiplicity of solutions for a class of Helmholtz systems

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This paper establishes a dual variational framework for studying the existence and multiplicity of solutions to a class of Helmholtz systems. We first focus on a compact nonlinear potential and prove the existence and multiplicity of solutions via the symmetric Mountain Pass theorem. Next, we investigate the periodic nonlinear potential case and prove a nonvanishing theorem to recover the loss of compactness. We then demonstrate the existence of solutions using the Nehari manifold method and the multiplicity of solutions using the pseudoindex theory. Additionally, the regularity of solutions is given.