Fredholm quasilinear manifolds and degree of a Fredholm quasilinear mapping between them
The author continues his study of quasilinear Fredholm mappings [the author, Turk. J. Math. 24, No. 1, 1--14 (2000; Zbl 0971.47045); ibid. 28, No. 3, 205--215 (2004; Zbl 1077.58003); ibid. 30, No. 2, 129--138 (2006; Zbl 1108.55004)] with the intention to generalize \textit{A. I. Shnirel'man}'s results [Mat. Sb., N. Ser. 89(131), 366--389 (1972; Zbl 0249.55003)]. Let \(M\) and \(N\) be compact smooth manifolds of dimension \(m<n\), where \(N\) is assumed to be closed. The author's aim is to prove that the Sobolev manifold \(H_s(M,N)\) is an FQL-manifold as defined by Šnirel'man [loc.\,cit.]. In order to achieve this, he views \(N\) as a submanifold of \(\mathbb{R}^{2n+1}\). Let then \(u\in H_s(M,\mathbb{R}^{2n+1})\) with \(\|u\|_s<j\). Alas, there are some puzzling facts in the proof: On p.\,677, line 5, the author estimates \(\|u\|_{C'}\) which is nowhere defined. Maybe it is \(\|u\|_{C^1}\) which also appears later, but would that not require a condition on \(s\)? On the following page, on line 1, he estimates \(\|u\|_{C^1}\leq K\|u\|_s\) ``for sufficiently large \(s\)'' -- but \(s\) is fixed and might not be ``sufficiently large''. Moreover, on the same page, line 4, he claims that (with \(K\) the constant from above and \(d\) the metric on \(M\)), for any \(u\in H_s(M,\mathbb{R}^{2n+1})\) with \(\|u\|_s<j\) and any \(x',x''\in M\), we have \(\|u\|_s<Kjd(x',x'')\), which is absurd even if one had \(\leq\) instead of \(<\). The author then proceeds to the case \(m\geq n\) and he finally tries to define a degree function for FSQL-mappings (cf.\ the first of the author's papers listed above for the definition) \(f:X\to X'\) with FQL-manifolds \(X\) and \(X'\) embedded in Banach spaces \(E_1\) and \(E_2\) satisfying the awkward hypothesis \(\|x\|_1\leq \|f(x)\|_2\) whenever \(x\in X\). This degree has the property that \(\deg(f, \cdot)\) is constant on \(X'\). As a matter of fact, the degree is defined only up to sign, so it is more or less a mod 2-degree. The interested reader should consult the articles by \textit{P. M. Fitzpatrick, J. Pejsachowicz} and \textit{P. J. Rabier} [J. Funct. Anal. 124, No. 1, 1--39 (1994; Zbl 0802.47056)], \textit{P. Benevieri} and \textit{M. Furi} [Ann. Sci. Math. Qué. 22, No. 2, 131--148 (1998; Zbl 1058.58502); Topol. Methods Nonlinear Anal. 16, No. 2, 279--306 (2000; Zbl 1007.47026)], and \textit{S.-G. Wang} [Mich. Math. J. 53, No. 2, 419--428 (2005; Zbl 1093.58003)] in order to learn how to keep track of orientations in order to get an integer-valued degree. The English of the article is rather unorthodox.
scientific article; zbMATH DE number 6016083