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Open Access Research

Existence results for classes of infinite semipositone problems

Jerome Goddard1, Eun Kyoung Lee2, Lakshmi Sankar3 and R Shivaji4*

Author Affiliations

1 Department of Mathematics, Auburn University Montgomery, Montgomery, AL, 36124, USA

2 Department of Mathematics Education, Pusan National University, Busan, 609-735, Korea

3 Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

4 Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, NC, 27412, USA

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Boundary Value Problems 2013, 2013:97  doi:10.1186/1687-2770-2013-97

Published: 19 April 2013

Abstract

We consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>, Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M9">View MathML</a>. Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M10">View MathML</a> is presented.