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Existence of an unbounded branch of the set of solutions for Neumann problems involving the p(x)-Laplacian

Byung-Hoon Hwang12, Seung Dae Lee2 and Yun-Ho Kim2*

Author Affiliations

1 Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea

2 Department of Mathematics Education, Sangmyung University, Seoul, 110-743, Republic of Korea

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Boundary Value Problems 2014, 2014:92  doi:10.1186/1687-2770-2014-92

Published: 6 May 2014

Abstract

We are concerned with the following nonlinear problem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M2">View MathML</a> in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M3">View MathML</a> on Ω, which is subject to a Neumann boundary condition, provided that μ is not an eigenvalue of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian. The aim of this paper is to study the structure of the set of solutions for the degenerate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian Neumann problems by applying a bifurcation result for nonlinear operator equations.

MSC: 35B32, 35D30, 35J70, 47J10, 47J15.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/92/mathml/M1">View MathML</a>-Laplacian; weighted variable exponent Lebesgue-Sobolev spaces; Neumann boundary condition; eigenvalue