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Two weak solutions for perturbed non-local fractional equations

In the present paper, we consider problems modeled by the following non-local fractional equation:where is fixed, is the fractional Laplace operator, are real parameters, is an open bounded subset of () with Lipschitz boundary , and are two suitable Carathéodory functions. By using variational methods, we prove the existence of at least two weak solutions for such problems for certain values of the parameters.