Bifurcation and stability of positive solutions of a two-point boundary value problem

Abstract

We consider the existence of multiple positive solutions of a nonlinear two-point boundary value problem by modifying a "time map" technique introduced by J. Smoller and A. Wasserman. We count the number of positive solutions and find their Conley indices and thus determine their stabilities.1991 Mathematics subject classification (Amer. Math. Soc): 58 F 14, 34 B 15.

Classification of bifurcation diagrams of a p-Laplacian Dirichlet problem with examples

“…The next Lemma 3.1 for 1 < p 2 and Lemma 3.2 for p > 2 are of independent interest. In particular, Lemma 3.1 extends[19, Theorem] for (3.1) from p = 2 to 1 < p 2.…”

Stationary patterns to diffusion problems

“…This permits us to elucidate the ro( le of the relative values of the multiplicities of such zeros u G in the arising of the corresponding transition layers therefore sharpening previous results in the subject [14]. [21]). We are producing there an &ad hoc' approach to the degeneracy issue which seems more general than the one in [16] and is directly based upon the knowledge of the asymptotic behaviour of ¹(u H ).…”

“…Reciprocally, families of solutions +u H , with arbitrary pre-"xed limit positions of such intervals can be constructed by a suitable choice of convergent sequences d G ( )/( and d G ( )/( in (21).…”

Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial

“…These and by (7), we obtain that θ (β 2 ) = −β 2 f (β 2 ) ≤ 0 and θ (β 3 ) = −β 3 f (β 3 ) > 0. So we obtain β 3 > C 2 by (10). Since β3 β1 f (u)du > 0 and by (5), θ(β 3 ) = 2F (β 3 ) > 2F (β 1 ) > 0.…”