Open Access Research

On the Fučík spectrum of the scalar p-Laplacian with indefinite integrable weights

Wei Chen1, Jifeng Chu1, Ping Yan2* and Meirong Zhang2

Author Affiliations

1 Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, People’s Republic of China

2 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

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Boundary Value Problems 2014, 2014:10  doi:10.1186/1687-2770-2014-10

Published: 9 January 2014

Abstract

In this paper, we study the structure of the Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> of Dirichlet and Neumann problems for the scalar p-Laplacian with indefinite weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. Besides the trivial horizontal lines and vertical lines, it will be shown that, confined to each quadrant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> is made up of zero, an odd number of, or a double sequence of hyperbolic like curves. These hyperbolic like curves are continuous and strictly monotonic, and they have horizontal and vertical asymptotic lines. The number of the hyperbolic like curves is determined by the Dirichlet and Neumann half-eigenvalues of the p-Laplacian with weights a and b. The asymptotic lines will be estimated by using Sturm-Liouville eigenvalues of the p-Laplacian with a weight a or b.

MSC: 34B09, 34B15, 34L05.

Keywords:
indefinite weights; p-Laplacian; Fučík spectrum; spectral structure