Fredholm and properness properties of quasilinear elliptic operators on \(\mathbb{R}^N\) (Q2770430)

This paper covers a fundamental step for the application of topological degree techniques to existence, multiplicity and bifurcation problems for solutions of second order quasilinear elliptic equations on whole of \(\mathbb{R}^N\). Since the embedding of \(W^{2,p}(\mathbb{R}^N)\) into \(L^p(\mathbb{R}^N)\) is not compact, the Leray-Schauder degree cannot be used here. One of the possible approaches is based on topological degree for \(C^1\)-Fredholm maps of index 0. In order to make use of this tool, it is necessary to have practical criteria, stated in terms of the coefficients of the equation, which allow to establish if the induced operator \(F\) in relevant function spaces is proper on closed and bounded subsets of its domain and if its Frechet derivative \(DF(u)\) is Fredholm. First of all the authors prove that, under very general conditions on the ``coefficients'' of the quasilinear equation in \(DF(n)\) is (left) semi-Fredholm at any point \(u\in W^{2,p}(\mathbb{R}^N)\) if and only if it is semi-Fredholm at one given point. Now, assuming this, they reduce the properness problem to a weaker one, of whether a bounded sequence \(u_n\) with \(F(u_n)\) convergent vanishes uniformly at infinity in the \(C^1\)-sense. Then, sufficient conditions for the vanishing at infinity are described in terms of an alternative typical of problems with lack of compactness due to invariance under translations (theorem 4.6). In a separate section they show that semi-Fredholmness of the Frechet derivative at some point is necessary for the properness of \(F\) on closed bounded sets. In the final part of the paper the authors apply the above results to obtain properness of \(F\) in terms of the asymptotic behavior of coefficients of the equation. Assuming that each coefficient converges to an \(N\)-periodic function of \(x\in\mathbb{R}^N\) as \(x\) goes to infinity, sufficient conditions are given in terms of the resulting asymptotic operator \(F^\infty\). For example, if \(F\) is semi-Fredholm, properness on closed bounded sets for \(F\) follows from the uniqueness of solutions of the equation \(F^\infty(u)=0\). Similar results are also obtained in terms of the asymptotic behavior of coefficients along rays through origin. On its turn, sufficient conditions for the uniqueness in the problem \(F^\infty(u)=0\), can be obtained either from the maximum principle or, in the variational case, from the Pohozaev type identities [cf. \textit{P. J. Rabier} and \textit{C. A. Stuart}, J. Differ. Equations 168, No. 1, 199-234 (2000; Zbl 0965.35046)]. Under some extra assumptions, the index of \(DF(u)\) was also determined by the authors using the results on spectral theory of the Laplacian perturbed by an \(N\)-periodic potential [cf. \textit{P. J. Rabier} and \textit{C. A. Stuart}, J.Differ. Integral Equ., 13, No. 10-12, 1429-1444 (2000; Zbl 0989.47036)]. Compared to other work of similar, technical nature I found this paper very clear and readable.