Curves of equiharmonic solutions and solvability of elliptic systems. (English) Zbl 1249.35110

Summary: We study solutions of the system \[ \Delta u+kf(v)=h_1(x),\; x\in \Omega ,\; u=0\; \text{ for }x\in \partial \Omega , \]
\[ \Delta v+kg(u)=h_2(x),\; x\in \Omega ,\; v=0\; \text{ for }x\in \partial \Omega \] on a bounded smooth domain \(\Omega \in \mathbb {R}^n\) with given functions \(f(t),g(t)\in C^2(\mathbb {R})\), 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 [J. Math. Mech. 19, 609–623 (1970; Zbl 0193.39203)] type, and sufficient conditions for existence of solutions of D. G. de Figueiredo and W.-M. Ni [Nonlinear Anal., Theory, Methods Appl. 3, 629–634 (1979; Zbl 0429.35035)] type. Our approach is constructive, and computationally efficient.


35J61 Semilinear elliptic equations
35J47 Second-order elliptic systems
35J57 Boundary value problems for second-order elliptic systems