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May  2009, 8(3): 999-1018. doi: 10.3934/cpaa.2009.8.999

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, Russian Federation

Received  June 2008 Revised  October 2008 Published  February 2009

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: fibering method, nonlinear Neumann boundary condition, Nonlinear partial differential equation, positive solution, bifurcation..
Mathematics Subject Classification: Primary: 35J65, 35P30; Secondary: 35J2.
Citation: Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999
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