  In this paper, we investigate the existence of nontrivial weak solutions to a
class of elliptic equations ($\mathscr{P}$) involving a general nonlocal
integrodifferential operator $\mathscr{L}_{\mathcal{A}K}$, two real parameters,
and two weight functions, which can be sign-changing. Considering different
situations concerning the growth of the nonlinearities involved in the problem
($\mathscr{P}$), we prove the existence of two nontrivial distinct solutions
and the existence of a continuous family of eigenvalues. The proofs of the main
results are based on ground state solutions using the Nehari method, Ekelands
variational principle, and the direct method of the calculus of variations. The
difficulties arise from the fact that the operator $\mathscr{L}_{\mathcal{A}K}$
is nonhomogeneous and the nonlinear term is undefined.
