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Existence of solutions for a general quasilinear elliptic system via perturbation method

Yujuan Jiao1*, Shengmao Fu2 and Yanli Wang3

Author Affiliations

1 College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730124, P.R. China

2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

3 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China

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Boundary Value Problems 2013, 2013:219  doi:10.1186/1687-2770-2013-219

Published: 7 November 2013

Abstract

In this paper, we consider the following quasilinear elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M8">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/219/mathml/M10">View MathML</a>) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC: 35J60, 35B33.

Keywords:
quasilinear elliptic system; positive solution; negative solution; perturbation method