Curves of equiharmonic solutions and solvability of elliptic systems

Abstract

We study solutions of the system \begin{eqnarray} \nonumber %26 \Delta u + k f(v)=h_1(x), \ x \in \Omega, \ u=0 \ \mbox{for $x \in \partial \Omega$} \\ \nonumber %26 \Delta v+kg(u)=h_2(x), \ x \in \Omega, \ v=0 \ \mbox{for $x \in \partial \Omega$} \nonumber \end{eqnarray} on a bounded smooth domain $\Omega \subset R^n$, with given functions $f(t)$, $g(t) \in C^2(R)$, and $h_1(x)$, $h_2(x) \in L^2(\Omega)$. When the parameter $k=0$, the problem is linear, and uniquely solvable. We continue the solutions in $k$ on curves of equiharmonic solutions. We show that in the absence of resonance the problem is solvable for any $h_1(x)$, $h_2(x) \in L^2(\Omega) $, while in the case of resonance we develop necessary and sufficient conditions for existence of solutions of E.M. Landesman and A.C. Lazer [12] type, and sufficient conditions for existence of solutions of D.G. de Figueiredo and W.-M. Ni [7] type. Our approach is constructive, and computationally efficient.