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Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent

Yuanze Wu* and Yisheng Huang

Author Affiliations

Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, P.R. China

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Boundary Value Problems 2013, 2013:149  doi:10.1186/1687-2770-2013-149

Published: 19 June 2013

Abstract

In this paper, we study the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M2">View MathML</a> is a smooth bounded domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M4">View MathML</a> is the p-Laplacian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M5">View MathML</a> is the critical Sobolev exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a> is a parameter. By establishing a new deformation lemma, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M7">View MathML</a>, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/149/mathml/M6">View MathML</a>, this problem has infinitely many sign-changing solutions, which extends the results obtained in (Cao et al. in J. Funct. Anal. 262: 2861-2902, 2012; Schechter and Zou in Arch. Ration. Mech. Anal. 197: 337-356, 2010).