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In this work, we are interested in the existence and multiplicity of nontrivial solutions for a class of elliptic problems. The first problem deals with the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator with variable exponent, two real parameters, and two weight functions, which can be sign-changing in a smooth bounded domain. The second problem deals with the existence and multiplicity of weak solutions involving the same operator, variable exponents without Ambrosetti and Rabinowitz type growth conditions and a positive real parameter in a smooth bounded domain.