Crandall-Rabinowitz Type Bifurcation for Non-differentiable Perturbations of Smooth Mappings

Abstract

We consider abstract equations of the type $F(\lambda ,u)=\tau G(\tau ,\lambda ,u)$, where $\lambda$ is a bifurcation parameter and $\tau$ is a perturbation parameter. We suppose that $F(\lambda ,0)=G(\tau ,\lambda ,0)=0$ for all $\lambda$ and $\tau$, F is smooth and the unperturbed equation $F(\lambda ,u)=0$ describes a Crandall-Rabinowitz bifurcation in $\lambda =0$, that is, two half-branches of nontrivial solutions bifurcate from the trivial solution in $\lambda =0$. Concerning G, we suppose only a certain Lipschitz condition; in particular, G is allowed to be non-differentiable. We show that for fixed small $\tau \ne 0$ there exist also two half-branches of nontrivial solutions to the perturbed equation, but they bifurcate from the trivial solution in two bifurcation points, which are different, in general. Moreover, we determine the bifurcation directions of those two half-branches, and we describe, asymptotically as $\tau \rightarrow 0$, how the bifurcation points depend on $\tau$. Finally, we present applications to boundary value problems for quasilinear elliptic equations and for reaction-diffusion systems, both with small non-differentiable terms.

*The paper is dedicated to Bernold Fiedler’s 60th birthday.

Financial support by the Czech Academy of Sciences (RVO:67985840) is gratefully acknowledged by the second author. The third author has been supported by the Grant 13-00863S of the Grant Agency of the Czech Republic and by RVO:67985840. The fourth author was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS16/239/OHK4/3T/14.