  The main aim of the paper is to prove a fountain theorem without assuming the
$\tau$-upper semicontinuity condition on the variational functional. Using this
improved fountain theorem, we may deal with more general strongly indefinite
elliptic problems with various sign-changing nonlinear terms. As an
application, we obtain infinitely many solutions for a semilinear Schr\"odinger
equation with strongly indefinite structure and sign-changing nonlinearity.
