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The existence of positive solutions to an elliptic system with nonlinear boundary conditions

Chunhua Wang* and Jing Yang

Author Affiliations

School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, P.R. China

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Boundary Value Problems 2013, 2013:159  doi:10.1186/1687-2770-2013-159

Published: 1 July 2013

Abstract

In this paper, we consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M1">View MathML</a>

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3">View MathML</a>) with smooth boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4">View MathML</a> is the outer normal derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5">View MathML</a> are positive and continuous functions. Under certain assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7">View MathML</a>, but without the usual (AR) condition, we prove that the problem has at least one positive strong pair solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a> (see Definition 1.4 below) by applying a linking theorem for strong indefinite functional.

MSC: 35A01, 35J20, 35J25.

Keywords:
fractional Sobolev spaces; linking theorem; nonlinear boundary conditions; strong solution