  In this paper we consider a class of logarithmic Schr\"{o}dinger equations
with a potential which may change sign. When the potential is coercive, we
obtain infinitely many solutions by adapting some arguments of the Fountain
theorem, and in the case of bounded potential we obtain a ground state
solution, i.e. a nontrivial solution with least possible energy. The functional
corresponding to the problem is the sum of a smooth and a convex lower
semicontinuous term.
