On resonant elliptic systems with rapidly rotating nonlinearities. (English) Zbl 1274.35103

The authors consider a nonlinear elliptic system with Neumann boundary conditions of the form \[ \Delta u+g(u)=p \text{ in } \Omega ,\quad \frac {\partial u}{\partial \nu }=0\;\text{ on } \partial \Omega , \tag{1} \] where \(\Omega \subset \mathbb {R}^d\) is a bounded domain, \(g\: \mathbb {R}^n \to \mathbb {R}^n\) is a continuous vector-valued function and the continuous forcing term \(p : \bar \Omega \to \mathbb {R}^n\) has mean value zero.
For the scalar case \(n = 1\) the well-known Landesman-Lazer theorem yields the existence of a solution if \(g\) satisfies \(g(+\infty )\cdot g(-\infty ) < 0\), where \(\displaystyle g(\pm \infty ) := \lim _{s \to \pm \infty } g(s)\). Generalizations of this result to systems were given by L. Nirenberg [Contrib. nonlin. functional Analysis, Proc. Sympos. Univ. Wisconsin, Madison, 1–9 (1971; Zbl 0267.47034)] and D. Ruiz and J. R. Ward [Discrete Contin. Dyn. Syst. 11, No. 2–3, 337–350 (2004; Zbl 1063.34034)], by imposing a condition which excludes “rapidly rotating” vector-valued nonlinearities. Indeed, R. Ortega and L. A. Sánchez [Bull. Lond. Math. Soc. 34, No. 3, 308–318 (2002; Zbl 1041.34003)] gave an example which shows that such equations with rapidly rotating nonlinearities may have no solutions. In their main theorem the authors give an existence result for system (1) under conditions which allow certain rapidly rotating nonlinearities \(g\).


35J57 Boundary value problems for second-order elliptic systems
35J60 Nonlinear elliptic equations