Asymptotic bifurcation and second order elliptic equations on \( R^{N} \)

Abstract

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation G(u) = λu, where G acts between real Hilbert spaces and λ ∈ R, and then for square-integrable solutions of a second order non-linear elliptic equation on R N . The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the H-asymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the H-asymptotic derivative if it is too close to the essential spectrum.

Bifurcation from infinity for an asymptotically linear Schrödinger equation

“…We show that if λ 0 is an isolated eigenvalue of odd multiplicity for L and if the distance dist (λ 0 , σ e (L)) from λ 0 to the essential spectrum of L is larger than the asymptotic Lipschitz constant of N (introduced in Definition 2.1(ii)), then λ 0 is an asymptotic bifurcation point for the equation Here we have assumed for notational simplicity that the asymptotic derivative N ′ (∞) of N is 0, see Theorem 1.1 for the full statement. This theorem slightly extends some results in [20,21] where the distance condition on λ 0 was somewhat stronger. If N is the gradient of a C 1 -functional and λ 0 is an isolated eigenvalue of finite (not necessarily odd) multiplicity, we show that under an additional hypothesis λ 0 is an asymptotic bifurcation point for (1.2).…”

“…We consider the asymptotically linear Schrödinger equation (1.1) and show that if λ0 is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence (un, λn) of solutions such that un → ∞ and λn → λ0. Our results extend those by Stuart [21]. We use degree theory if the multiplicity of λ0 is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.…”

“…As we shall see, if the distance condition is satisfied, then one can find an orthogonal decomposition E = Z ⊕ W , where dim Z < ∞, such that writing u = z + w ∈ Z ⊕ W , it is possible to use the contraction mapping principle in order to express w as a function of z and λ. Although one may think this is only a technical condition, it has been shown by Stuart [21,Section 5.2] that there exist examples where asymptotic bifurcation does not occur at eigenvalues of odd multiplicity (and in Section 5.3 there one finds an example where asymptotic bifurcation occurs when λ 0 is not an eigenvalue). So the above condition, or some other, is needed.…”

Bifurcation at isolated singular points of the Hadamard derivative

“…We have chosen to deal with C rather than C in this work because it has the advantage that, in applications to asymptotic bifurcation via inversion [19], its image under inversion is connected.…”

Bifurcation without Fréchet differentiability at the trivial solution

“…A more systematic investigation in an abstract variational setting was begun in joint work with Gilles Evéquoz where we also investigated new applications involving nonlinear elliptic equations on (see also ) and degenerate elliptic equations on bounded domains . New impetus has been provided by the realization that the study of asymptotic bifurcation for stationary Schrödinger equations through inversion leads to a bifurcation problem where Fréchet differentiability is lacking . The abstract results obtained in are based on a variational method in which the existence of non‐trivial solutions is proved first and then bifurcation is established ‘a postiori ’.…”