Positivity of global branches of fully nonlinear elliptic boundary value problems

Healey, Timothy J.; Kielhöfer, Hansjörg

doi:10.1090/S0002-9939-1992-1091182-5

Positivity of global branches of fully nonlinear elliptic boundary value problems
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by Timothy J. Healey and Hansjörg Kielhöfer PDF

Proc. Amer. Math. Soc. 115 (1992), 1031-1036 Request permission

Abstract:

We consider a bifurcation problem for a general class of fully nonlinear, second-order elliptic equations on a regular bounded domain in ${\mathbb {R}^n}$ and subject to homogeneous Dirichlet boundary data. We assume that the linearized problem about the trivial solution possesses a positive solution for at least one isolated parameter value. With no other growth or sign conditions imposed upon the nonlinearity, we establish the existence of a global branch of nontrivial positive solutions. Moreover, if there is only one such isolated value of the parameter, we deduce that the branch of positive solutions is unbounded.

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