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

Korman, Philip

Differ. Integral Equ. 24, No. 11-12, 1147-1162 (2011).

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.

MSC:

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

Keywords:

elliptic system; equiharmonic solution; existence

Citations:

Zbl 0193.39203; Zbl 0429.35035