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We are concerned with $T$-periodic solutions of nonautonomous parabolicproblem of the form $u_t = \Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \in\mathbb{R}^N$, with $V \in L^\infty (\mathbb{R}^N)+L^p(\mathbb{R}^N)$, $p \geqN$ 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 ${\calN}:=\mathrm{Ker} (\Delta + V) \neq \{0\}$ and $f$ is bounded by asquare-integrable function. We derive a formula for the fixed point index ofthe associated translation along trajectories operator in terms of the Brouwertopological 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 andcontinuation techniques we show that Landesman-Lazer type conditions imply theexistence of $T$-periodic solutions.

A Landesman-Lazer type result for periodic parabolic problems on $\mathbb{R}^N$ at resonance