Opuscula Math. 32, no. 1 (2012), 91-113
http://dx.doi.org/10.7494/OpMath.2012.32.1.91

Opuscula Mathematica

# On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half space

Zagharide Zine El Abidine

Abstract. We study the existence of positive continuous solutions of the nonlinear polyharmonic system $$(-\Delta)^m u + \lambda q g(v) = 0$$; $$(-\Delta)^m v + \mu p f(u) = 0$$ in the half space $$\mathbb{R}^n_+:=\{x = (x_1,...,x_n) \in \mathbb{R}^n : x_n \gt 0\}$$, where $$m \geq 1$$ and $$n \gt 2m$$. The nonlinear term is required to satisfy some conditions related to the Kato class $$K^{\infty}_{m,n}(\mathbb{R}^n_+)$$. Our arguments are based on potential theory tools associated to $$(-\Delta)^m$$ and properties of functions belonging to $$K^{\infty}_{m,n}(\mathbb{R}^n_+)$$.

Keywords: polyharmonic elliptic system, Green function, Kato class, positive continuous solution, Schauder fixed point theorem.

Mathematics Subject Classification: 34B27, 35J40.

Full text (pdf)