Exact multiplicity of solutions and S-shaped bifurcation curve for a class of semilinear elliptic equations

Abstract

The set of steady state solutions to a reaction-diffusion equation modeling an autocatalytic chemical reaction is completely determined, when the reactor has spherical geometry, and the spatial dimension is n = 1 or 2 for any reaction order, or n 3 for subcritical reaction order. Bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states.

The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions

“…4, indicates that the upper and lower branches are locally stable and the middle branch unstable. Such bifurcation of solutions or behavior, common known as hysterisis, is present in many biological or physical systems and equations that exhibit solutions of this nature have been studied extensively by various authors (see, for examples, Du [17], Crandall and Rabinowitz [12], Wang [27] and Zhao, Wang, and Shi [32] and the references therein).…”

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

“…We shall refer to such kind of component as a CS-shaped bifurcation component. To the best of our knowledge, results stating the existence of S-shaped components in the positive solutions set of logistic equations are not known in the literature, unless in the one-dimensional case (see [4]) and in the radial case (see [18]). Furthermore, the existence of CS-shaped components seems not to have been observed.…”

Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications

“…This problem arises from an autocatalytic chemical reaction. We refer to [8,24] for more background of this problem and its generalizations. It is easy to see that, for k ∈ (0, 1), there exists a positive number β k < 1 which is the unique positive zero of the cubic polynomial f k (u) ≡ k 2 − k 3 + (2k − 3k 2 )u + (1 − 3k)u 2 − u 3 such that f k satisfies f k (0) = k 2 − k 3 > 0, f k (u) > 0 on (0, β k ) and f k (β k ) = 0.…”