  We establish the existence of an entire solution for a class of stationary
Schr\"{o}dinger equations with subcritical discontinuous nonlinearity and lower
bounded potential that blows-up at infinity. The abstract framework is related
to Lebesgue-Sobolev spaces with variable exponent. The proof is based on the
critical point theory in the sense of Clarke and we apply Chang's version of
the Mountain Pass Lemma without the Palais-Smale condition for locally
Lipschitz functionals. Our result generalizes in a nonsmooth framework a result
of Rabinowitz on the existence of ground-state solutions of the nonlinear
Schr\"{o}dinger equation.
