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We consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

On a class of parametric Neumann problems with indefinite and unbounded potential