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A Landesman-Lazer type result for periodic parabolic problems on \(\mathbb R^N\) at resonance. (English) Zbl 1328.35107

Summary: 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 \(V\in L^\infty(\mathbb R^N)+L^p(\mathbb R^N)\), where \(2<p<\infty\) if \(N=1,2\) and \(N\leq p<\infty\) for \(N\geq 3\) 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 \(\mathcal N:=\mathrm{Ker}(\Delta+V)\neq \{0\}\) and \(f\) is bounded. 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 averaging of the mapping \(\hat{f}:\mathcal N\to\mathcal N\), \(\hat{f}\) 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.

MSC:

35K58 Semilinear parabolic equations
35B10 Periodic solutions to PDEs
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