Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem

Vladimir Lubyshev

doi:10.3934/cpaa.2009.8.999

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2009, Volume 8, Issue 3: 999-1018. Doi: 10.3934/cpaa.2009.8.999

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Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem

Vladimir Lubyshev^1,

1.
Bashkir State University, Department of Mathematics, Ufa, 450000

Received: May 31, 2008

Revised: September 30, 2008

Published: April 2009

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Abstract

We study positive solutions of an elliptic problem with indefinite in sign nonlinear Neumann boundary condition that depends on a real parameter, $\lambda$. We find precise range, $I$, of those $\lambda$'s for which our problem possesses a positive solution, prove that $\lambda^$∗ = sup $I$ is a bifurcation point, and exhibit explicit max-min procedure for computing $\lambda^$∗. We also obtain some properties of the set of solutions.

Keywords:

Nonlinear partial differential equation,
nonlinear Neumann boundary condition,
positive solution,
fibering method,
bifurcation.

Mathematics Subject Classification: Primary: 35J65, 35P30; Secondary: 35J20.

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