Nonlinear perturbations of linear elliptic systems at resonance

Abstract:

We consider a semilinear system \begin{align*} \Delta u&+ \lambda v+b_1(v)=f(x),\;\; x \in \Omega ,\quad \;\;\; u=0 \mbox {\ \ \ for $x \in \partial \Omega $} ,\\ \Delta v&+\frac {\lambda ^2 _1}{\lambda } u+b_2(u) =g(x),\;\; x \in \Omega ,\quad v=0 \mbox {\ \ \ for $x \in \partial \Omega $}, \end{align*} whose linear part is at resonance. Here $\lambda >0$ and the functions $b_1(t)$ and $b_2(t)$ are bounded and continuous. Assuming that $tb_i(t)>0$ for all $t \in R$, $i=1,2$, and that the first harmonics of $f(x)$ and $g(x)$ lie on a certain straight line, we prove the existence of solutions. This extends a similar result for one equation, due to D.G. de Figueiredo and W.-M. Ni.