Preface The p-Laplacian operator Δp u div ` |∇u|p´2 ∇u ˘ , p P p1, 8q arises in non-Newtonian fluid flows, turbulent filtration in porous media, plasticity theory, rheology, glaciology, and in many other application areas see, e.g., Esteban and azquez [48] and Padial, Tak´c, and Tello [90]. Prob- lems involving the p-Laplacian have been studied extensively in the literature during the last fifty years. However, only a few papers have used Morse the- oretic methods to study such problems see, e.g., Vannella [130], Cingolani and Vannella [29, 31], Dancer and Perera [40], Liu and Su [74], Jiu and Su [58], Perera [98, 99, 100], Bartsch and Liu [15], Jiang [57], Liu and Li [75], Ayoujil and El Amrouss [10, 11, 12], Cingolani and Degiovanni [30], Guo and Liu [55], Liu and Liu [73], Degiovanni and Lancelotti [43, 44], Liu and Geng [70], Tanaka [129], and Fang and Liu [50]. The purpose of this monograph is to fill this gap in the literature by presenting a Morse theo- retic study of a very general class of homogeneous operators that includes the p-Laplacian as a special case. Infinite dimensional Morse theory has been used extensively in the lit- erature to study semilinear problems (see, e.g., Chang [28] or Mawhin and Willem [81]). In this theory the behavior of a C1-functional defined on a Banach space near one of its isolated critical points is described by its critical groups, and there are standard tools for computing these groups for the variational functional associated with a semilinear problem. They include the Morse and splitting lemmas, the shifting theorem, and various linking and local linking theorems based on eigenspaces that give critical points with nontrivial critical groups. Unfortunately, none of them apply to quasilinear problems where the Euler functional is no longer defined on a Hilbert space or is C2 and there are no eigenspaces to work with. We will systematically develop alternative tools, such as nonlinear linking and local linking theories, in order to effectively apply Morse theory to such problems. A complete description of the spectrum of a quasilinear operator such as the p-Laplacian is in general not available. Unbounded sequences of eigen- values can be constructed using various minimax schemes, but it is generally not known whether they give a full list, and it is often unclear whether dif- ferent schemes give the same eigenvalues. The standard eigenvalue sequence based on the Krasnoselskii genus is not useful for obtaining nontrivial critical vii
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