Abstract
In the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points and intervals of the line trivial solutions, corresponding to the positive or negative eigenvalues of the linear problem. As applications of this result, these authors study the existence of nodal solutions for a class of nonlinear fourth-order eigenvalue problems with sign-changing weight. Moreover, they also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. In the present comment, we show that these papers of above authors contain serious errors and, therefore, unfortunately, the results of these works are not true. Note also that the authors used the results of the recent work by G. Dai which also contain gaps.
We want to point out that the assertions of the papers [1, 2] cannot be true, as they contradict classical results of mathematical analysis, since nonlinear eigenvalue problems of fourth-order arise in many applications, (see [3, 4] and the references therein).
In the works [1, 2], based on the spectral theory of [5], the authors establish the unilateral global bifurcation result about the continuum of solutions for the following fourth-order eigenvalue problem:where is a positive function [1] or sign-changing function [2] on , satisfies the Carathéodory condition, and . Let and(In the paper [1] also studied global bifurcation for nonlinearizable and half-linearizable eigenvalue problems of fourth-order). It is also assumed that the function is continuous and satisfies the following condition:uniformly for and on bounded sets.
By (3), the linearization of (1) at is the spectral problem
Let with the norm
Let and such that . We note that is a generalized simple zero if or . Otherwise, we note that is a generalized double zero. If there is no generalized double zero of , we note that is a nodal solution [1, 2].
The linear problem (4) is investigated in [5] (see also [1, Lemma ]) where, in particular, the following theorem is proved.
Theorem A. Let . The eigenvalue problem (4) has two sequences of simple real eigenvaluesand no other eigenvalues. Moreover, for each and each , the eigenfunction , corresponding to the eigenvalue , has exactly generalized simple zeros in .
Note that if is positive on then problem (4) has one sequence of positive eigenvalues (see [1]).
Let , , denote the set of functions in which have exactly generalized simple zeros in and are positive near and set , and .
Let be the closure of the set of nontrivial solution of problem (1).
One of the main results of the work [2] is the following theorem which plays an essential role in the study of problems considered in [1].
Theorem B ([2, Theorem ] (see also [1, Lemma ])). Assume that (3) holds and , then from each it bifurcates two distinct unbounded continua and of . Moreover, for , we have that
In [1, p. 2] and [2, p. 9401] the authors write that “Clearly, the sets , are disjoint and open in ” and further they use this assertion for the prove of Theorem B. It is obvious that these sets are disjoint, but they are not open in . Hence, the statements of the Theorem B and [1, Theorems and ] are not true.
Now we will show that for the set , are not open in . For simplicity, consider the case (this fact is shown for any similarly).
Let and . Note that , , and . We take a sufficiently small number and consider the function . Then we have . Next, we consider the following equations:The solution of the first equation is and the solution of the second equation is and . Hence is not a generalized zero of the function in , although is contained in neighborhood of the function in . This means that is not open in . Hence is also not open in .
In Section from [2] the authors established the Sturm type comparison theorem for fourth-order differential equations with sign-changing weight, which they used later in [2] and also in [1] (see [1, page 6, left column, line 6 from below]).
Lemma A (see [2, Lemma3.1]). Let for and . Also let be nontrivial solutions of the following differential equations:respectively. If has generalized simple zeros in , then has at least generalized simple zeros in .
The proof of Lemma A contain gaps. Now we demonstrate this fact. Let . It follows by (9) that
In the proof of Lemma A the authors claim that (see [2, p. 9403, formula ()]) “By simple computation, one has thatfor any constant .”
Multiplying the equations in (10) by and , respectively, and adding both sides we obtainIntegrating this relation from to , we haveFormula (13) shows that the formula (11) is not true, except in the case .
We show that the statement of theorem A ([1, Lemma ]) that the eigenfunction , corresponding to the eigenvalue , has exactly generalized simple zeros in is not true. The proof of this statement has been achieved as follows (see proof of Proposition3.3 from [5]). Let be the first zero of , in . Set . For any , let be the extension by zero of on . It is obvious that . By Definition2.1 from [5], we haveHence the restriction of in is a nonnegative solution of the following problem:In fact, the restriction of in is a classical solution of problem (15). Indeed, Proposition2.1 from [5] yields in is a classical solution of problem (1). Hence satisfies in .
It remains to show that . Let us choose . Substituting in (14) and integrating by parts we obtainSince is arbitrary, it follows from last equality that .
Note that this proof is not true. Indeed, if, for any , one lets be the extension by zero of on , then it does not follow that . By the embedding theorem (see [6]), we have with , and so . If we take a function such that , then by definition of it follows that . But, then again, by definition of we have . From this we conclude that the function has not a derivative at the point . Hence, by the embedding . Thus the equality (14) is not true for all .
Competing Interests
There are no competing interests related to this paper.