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

Abstract

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 $V\in L^{\infty }({\mathbb{R}}^N)+L^p({\mathbb{R}}^N)$, $p\ge N$ 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 $\mathcal{N}:=\mathrm{Ker}(\Delta +V)\ne \{0\}$ 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 $\hat{f}:\mathcal{N}\to \mathcal{N}$ being the restriction of $f$ to $\mathcal{N}$. By use of the formula and continuation techniques we show that Landesman-Lazer type conditions imply the existence of $T$-periodic solutions.

arXiv admin note: text overlap with arXiv:1404.0256