Elliptic equations involving nonlocal operator with exponent variable

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.