  We are concerned with $T$-periodic solutions of nonautonomous parabolic
problem of the form $u_t = \Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \in
\mathbb{R}^N$, with $V \in L^\infty (\mathbb{R}^N)+L^p(\mathbb{R}^N)$, $p \geq
N$ and $T$-periodic continuous perturbation $f:\mathbb{R}^N\times \mathbb{R}
\to \mathbb{R}$. The so-called resonant case is considered, i.e. when ${\cal
N}:=\mathrm{Ker} (\Delta + V) \neq \{0\}$ and $f$ is bounded by a
square-integrable function. We derive a formula for the fixed point index of
the associated translation along trajectories operator in terms of the Brouwer
topological degree of the time average mapping $\hat f: {\cal N}\to {\cal N}$
being the restriction of $f$ to ${\cal N}$. By use of the formula and
continuation techniques we show that Landesman-Lazer type conditions imply the
existence of $T$-periodic solutions.

Analysis of PDEs