Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 209, pp. 1-13.
Title: Range of semilinear operators for systems at resonance
Authors: Pablo Amster (Univ. de Buenos Aires, Argentina)
Mariel Paula Kuna (Univ. de Buenos Aires, Argentina)
Abstract:
For a vector function $u:\mathbb{R} \to \mathbb{R}^N $ we consider the system
$$\displaylines{
u''(t)+ \nabla G(u(t))= p(t)\cr
u(t)=u(t+T),
}$$
where $G: \mathbb{R}^N \to \mathbb{R}$ is a $C^1$ function.
We are interested in finding all possible T-periodic forcing terms
p(t) for which there is at least one solution.
In other words, we examine the
range of the semilinear operator
$S:H^2_{\rm per}\to L^2([0,T],\mathbb{R}^N)$ given by $Su= u''+ \nabla G(u)$,
where
$$
H^2_{\rm per}= \{ u\in H^2([0,T], \mathbb{R}^N);
u(0) - u(T) = u'(0)-u'(T)=0 \}.
$$
Writing
$p(t)= \overline{p} + \widetilde{p}(t)$, where
$\overline{p}:=\frac 1T\int_0^Tp(t)\, dt$,
we present several results concerning the topological structure of the set
$$
\mathcal{I}(\widetilde{p})=\{ \overline{p} \in \mathbb{R}^N;
\overline{p} + \widetilde{p}\in \operatorname{Im}(S)\}.
$$
Submitted October 7, 2011. Published November 27, 2012.
Math Subject Classifications: 34B15, 34L30.
Key Words: Resonant systems; semilinear operators; critical point theory.