Opuscula Mathematica
Opuscula Math. 32, no. 1 (), 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


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.
Cite this article as:
Zagharide Zine El Abidine, On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half space, Opuscula Math. 32, no. 1 (2012), 91-113, http://dx.doi.org/10.7494/OpMath.2012.32.1.91
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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