Correction to: Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons

This erratum corrects some errors which appear in [Dohnal T., Romani G., Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons, Nonlinear Differ. Equ. Appl. 28, 9 (2021).]. In particular, it replaces the (unsuitable) definition of isolatedness of the linear eigenvalue, from which the bifurcation occurs.


Introduction
In [1] we proved a result on bifurcation from simple isolated eigenvalues for a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter. The general result was then applied in the context of surface plasmon polaritons (SPPs). Although the proof of the main result is correct, the definition of the isolatedness for an eigenvalue is not suitable for our purposes and does not lead to the linearized operator being Fredholm. As a consequence, wrong assumptions were checked in the application to SPPs. In this erratum we provide the correct assumptions and verify that they are satisfied in our examples.
In doing so we also remove some minor errors and inconsistencies. Section 2 lists and explains all the necessary corrections. For readers' convenience we have incorporated all the corrections in the arXiv version of this paper [2]. Notation. The numbering of theorems and equations refers to the one used in [1]. New equations are numbered with (1'), (2'), etc.
1. The definition of the isolatedness of the linear eigenvalue ω 0 has to be modified. The original definition was the isolatedness of ω 0 in the complex ω−plane. However, for the Fredholm property of L(·, ω 0 ) (used in the proof of Theorem 2.1) the isolatedness of the zero eigenvalue of L(·, ω 0 ) (i.e. for ω = ω 0 fixed) is needed. The new assumptions at the beginning of Section 2 (to replace items (i) and (ii) on p. 3) are as follows (E1) ω 0 is algebraically simple in the sense that κ = 0 is an algebraically simple eigenvalue of the standard eigenvalue problem L(·, ω 0 )u = κu, i.e. ker(L(·, ω 0 ) 2 ) = ker(L(·, ω 0 )) = ϕ 0 , (E2) ω 0 is isolated in the sense that κ = 0 is an isolated eigenvalue of the problem L(·, ω 0 )u = κu. 2. We check these new assumptions (E1) and (E2) for the examples from Section 4.1.2 in Appendix A of this erratum. The text of the appendix is to be understood as an addition to Section 4.2. 3. Equation (7) in assumption (f4) has to be corrected (weakened to an estimate in L 2 ): 4. The statement of the main theorem should be formulated more clearly regarding the uniqueness statement: . ω 0 is an algebraically simple and isolated eigenvalue of L with eigenfunction ϕ 0 and that A, W and f satisfy assumptions (A1)-(f4). Let also τ ∈ (0, min{1, αβ}]. Then there is a unique branch bifurcating from (ω 0 , 0). There exists ε 0 > 0 s.t. for any ε ∈ (0, ε 0 ) the solution (ω, ϕ) normalized to satisfy ϕ, ϕ * 0 = ε α has the form (26): a constant C was missing in the last estimate:

Equation
Correction of a sign in the proof of Proposition 3.1. We have on p. 17, l. 2-3 and hence on p. 17, l. 11-12 8. Figure 1 (d) displays the plot ofd 1 and notd −1 . The caption for (d) and the text discussing this plot has to be modified accordingly. 9. In equation (62) the numerator 1 has to be replaced by 2π.
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Verification of (E1): ω 0 is simple. Because in (58)-(59) the constants A, B, C, D are unique (up to normalization of ϕ 0 ), it is clear that ker L(·, ω 0 ) = ϕ 0 , so 0 as eigenvalue of L(·, ω 0 ) is geometrically simple. To prove the algebraic simplicity, suppose by contradiction that there exists a Jordan chain associated to ω 0 . This means, there exists u ∈ D(A) (see (54)) such that Solving (1') explicitly using the variation of constants, one finds Note that T is singular since by our choice of d in (60). In order to find a contradiction and exclude the existence of a solution u ∈ D(A) of (1'), we now prove that b is not orthogonal to the kernel of T T . Standard computations show that ker T T is one-dimensional and given by The scalar product (b, p) then reads After evaluating the integrals for ϕ 0 given by (58)-(59), and some algebraic computations in which the identity e 2μd = (μ−λ+)(μ−λ−) (μ+λ+)(μ+λ−) is frequently used, we get Finally, we check that (b, p) is non-zero for the values of λ ± , μ, and d obtained numerically in Sect. 4.1.2. We obtain (b, p) ≈ −19.38 − 46.36i for Case 2 and (b, p) ≈ 0.82 + 1.58i for Case 3. Verification of (E2): ω 0 is isolated. For L(x, ω 0 ) := − d 2 dx 2 − W (x, ω 0 ) the essential spectrum is given by Since W ± (ω 0 ) ∈ C \ R in both Case 2 and Case 3, we have 0 / ∈ σ ess (L(·, ω 0 )). As the essential spectrum is closed, 0 is isolated from σ ess (L(·, ω 0 )).
Since κ → f (κ) is differentiable at 0, a necessary condition for f (κ j ) = 0 with κ j → 0 is f (0) = 0, that is By simple computations one gets This contradicts the fact that d > 0 and that λ ± are chosen with positive real part. As a consequence, we deduce that ω 0 is isolated in the sense of (E2).