Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

We establish the existence of an entire solution for a class of stationary Schrödinger 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 [P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. (ZAMP) 43 (1992) 270–291] on the existence of ground-state solutions of the nonlinear Schrödinger equation.