Bifurcation of Nonlinear Equations: I. Steady State Bifurcation (original) (raw)

We prove in this article some general steady state bifurcation theorem for a class of nonlinear eigenvalue problems, in the case where algebraic multiplicity of the eigenvalues of the linearized problem is even. These theorems provide an addition to the classical Krasnoselskii and Rabinowitz bifurcation theorems, which require the algebraic multiplicity of the eigenvalues is odd. For this purpose, we prove a spectral theorem for completely continuous fields, which can be considered as a generalized version of the classical Jordan matrix theorem and the Fredholm theorem for compact operators. An application to a system of second order elliptic equations is given as well.

Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators

Mediterranean Journal of Mathematics, 2022

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to L^{\infty }L∞,whichimpliesL ∞ , which impliesL∞,whichimpliesC^{1,\alpha }$$ C 1 , α smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

Bifurcation for an elliptic system coupled in the linear part

Nonlinear Analysis: Theory, Methods & Applications, 1999

We study linearly coupled elliptic systems without any assumption on the signs of the coefficients of the linear part. We derive first the Leray-Schauder degree of the associated linear system. Then we obtain sufficient conditions on the nonlinear terms for having bifurcation to coexistence states. Our results include the sublinear case.

Bifurcation of Nonlinear Equations: II. Dynamic Bifurcation

2004

We study in this article dynamic bifurcation of nonlinear evolution equations due to higher order nonlinear terms, focusing on detailed bifurcation behavior of nonlinear evolution equations in the cases where the algebraic multiplicity of the eigenvalues of the linearized problem is one or two.

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