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

TL;DR

This paper establishes a Landesman-Lazer type existence result for periodic solutions of resonant nonautonomous parabolic equations on ^N, using fixed point index and Brouwer degree techniques in the context of resonance.

Contribution

It derives a fixed point index formula at resonance for periodic parabolic problems and applies it to prove existence of solutions under Landesman-Lazer conditions.

Findings

Fixed point index formula in terms of Brouwer degree

Existence of periodic solutions at resonance

Application of continuation methods to nonlinear PDEs

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 $V \in L^\infty (\mathbb{R}^N)+L^p(\mathbb{R}^N)$, $p \geq N$ 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.