  In this paper, we study the existence and multiplicity of weak solutions for
a general class of elliptic equations (\mathscr{P}_{\lambda}) in a smooth
bounded domain, driven by a nonlocal integrodifferential operator
\mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving
variable exponents without Ambrosetti and Rabinowitz type growth conditions.
Using different versions of the Mountain Pass Theorem, as well as, the Fountain
Theorem and Dual Fountain Theorem with Cerami condition, we obtain the
existence of weak solutions for the problem (\mathscr{P}_{\lambda}) and we show
that the problem treated has at least one nontrivial solution for any parameter
\lambda>0 small enough as well as that the solution blows up, in the fractional
Sobolev norm, as \lambda \to 0. Moreover, for the case sublinear, by imposing
some additional hypotheses on the nonlinearity f(x,\cdot), by using a new
version of the symmetric Mountain Pass Theorem due to Kajikiya [36], we obtain
the existence of infinitely many weak solutions which tend to be zero, in the
fractional Sobolev norm, for any parameter \lambda>0. As far as we know, the
results of this paper are new in the literature.
