Nonlinear perturbations of linear elliptic systems at resonance
Philip Korman · Proceedings of the American Mathematical Society · 2011
We consider a semilinear system Δ u a m p ; + λ v + b 1 ( v ) = f ( x ) , x ∈ Ω , u = 0 \ \ \ \,\,for x ∈ ∂ Ω , Δ v a m p ; + λ 1 2 λ u + b 2 ( u ) = g ( x ) , x ∈ Ω , v = 0 \ \ \ \, for x ∈ ∂ Ω , \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 λ > 0 \lambda >0 and the functions b 1 ( t ) b_1(t) and b 2 ( t ) b_2(t) are bounded and continuous. Assuming that t b i ( t ) > 0 tb_i(t)>0 for all t ∈ R t \in R , i = 1 , 2 i=1,2 , and that the first harmonics of f ( x ) f(x) and g ( x ) 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.