New nonexistence results are obtained for entire bounded (either
from above or from below) weak solutions of wide classes of
quasilinear elliptic equations and inequalities. It should be
stressed that these solutions belong only locally to the
corresponding Sobolev spaces. Important examples of the situations
considered herein are the following: $\sum_{i=1}^n(a(x)|\nabla u|^{p-2}u_{x_i})_{x_i}= -|u|^{q-1}u$ , $\sum_{i=1}^n(a(x)|u_{x_i}|^{p-2}u_{x_i})_{x_i}= -|u|^{q-1}u$ , $\sum_{i=1}^n({a(x)|\nabla u|^{p-2}u_{x_i}}/{\sqrt{1+|nabla u|^2}})_{x_i}= -|u|^{q-1}u$ , where $n\geq 1,p > 1,q > 0
are fixed real numbers, and $a(x)$
is a nonnegative measurable locally bounded function. The methods involve the use of capacity theory in connection with special types of test
functions and new integral inequalities. Various results,
involving mainly classical solutions, are improved and/or extended
to the present cases.