Open Access Research

Positivity of the infimum eigenvalue for equations of p ( x ) -Laplace type in R N

In Hyoun Kim1 and Yun-Ho Kim2*

Author Affiliations

1 Department of Mathematics, Incheon National University, Incheon, 406-772, Republic of Korea

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

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Boundary Value Problems 2013, 2013:214  doi:10.1186/1687-2770-2013-214

Published: 2 October 2013

Abstract

We study the following elliptic equations with variable exponents

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M3">View MathML</a>

Under suitable conditions on ϕ and f, we show the existence of positivity of the infimum of all eigenvalues for the problem above, and then give an example to demonstrate our main result.

MSC: 35D30, 35J60, 35J92, 35P30, 47J10.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/214/mathml/M1">View MathML</a>-Laplacian; variable exponent Lebesgue-Sobolev spaces; weak solution; eigenvalue