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Multiplicity of positive solutions for eigenvalue problems of ( p , 2 ) -equations

Leszek Gasiński1* and Nikolaos S Papageorgiou2

Author affiliations

1 Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, ul. Łojasiewicza 6, Kraków, 30-348, Poland

2 Department of Mathematics, National Technical University, Zografou Campus, Athens, 15780, Greece

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Citation and License

Boundary Value Problems 2012, 2012:152  doi:10.1186/1687-2770-2012-152

Published: 28 December 2012

Abstract

We consider a nonlinear parametric equation driven by the sum of a p-Laplacian (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M2">View MathML</a>) and a Laplacian (a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M1">View MathML</a>-equation) with a Carathéodory reaction, which is strictly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M4">View MathML</a>-sublinear near +∞. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem for the nonlinear eigenvalue problem. So, we show that there is a critical parameter value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M5">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M6">View MathML</a> the problem has at least two positive solutions, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M7">View MathML</a>, then the problem has at least one positive solution and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/152/mathml/M8">View MathML</a>, it has no positive solutions.

MSC: 35J25, 35J92.

Keywords:
nonlinear regularity; tangency principle; p-Laplacian; bifurcation-type theorem; positive solutions