Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 354, Issue 8, Pages 3117-3154Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9947-02-03007-6
Keywords
semilinear elliptic equations; secondary bifurcations; global bifurcation diagrams; asymptotic behavior of solutions
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We consider a semilinear elliptic equation Deltau + lambdaf(u) = 0, x is an element of Omega, partial derivativeu/partial derivativen = 0, x is an element of partial derivativeOmega, where Omega is a rectangle (0, a) x (0, b) in R-2. For balanced and unbalanced f, we obtain partial descriptions of global bifurcation diagrams in (lambda, u) space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when lambda --> infinity. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.
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