## Differential and Integral Equations

### Curves of equiharmonic solutions and solvability of elliptic systems

Philip Korman

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 24, Number 11/12 (2011), 1147-1162.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356012881

Mathematical Reviews number (MathSciNet)
MR2866016

Zentralblatt MATH identifier
1249.35110

Subjects
Primary: 35J60: Nonlinear elliptic equations

#### Citation

Korman, Philip. Curves of equiharmonic solutions and solvability of elliptic systems. Differential Integral Equations 24 (2011), no. 11/12, 1147--1162. https://projecteuclid.org/euclid.die/1356012881