  We prove a general perturbation theorem that can be used to obtain pairs of
nontrivial solutions of a wide range of local and nonlocal nonhomogeneous
elliptic problems. Applications to critical $p$-Laplacian problems,
$p$-Laplacian problems with critical Hardy-Sobolev exponents, critical
fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are
given. Our results are new even in the semilinear case $p = 2$.
