Shapiro V.L. Singular Quasilinearity and Higher Eigenvalues

Providence: American Mathematical Society, 2001. — 174 p.This research monograph establishes many new results in the area of singular quasilinear elliptic and parabolic partial differential equations and
is mainly motivated by three of the author's previous papers, [Shi], [LfS], and [LgS]. The singularities that arise are described by weights which often can be associated with special functions like the Hermite polynomials and the associated Legendre functions.
Chapter 1 of this monograph deals with a singular quasilinear elliptic operator Q which is a perturbation of a singular elliptic operator L and contains results which improve upon those in [Shi] in three different directions. The first direction is to obtain results for Q at the higher eigenvalues of L. In particular, Theorem 1 establishes a quasilinear analogue of the Fredholm alternative for Q at specific higher eigenvalues of L; Theorem 4, motivated by [BdF] , gives a double resonance result for Q at all the eigenvalues of L; and Theorem 6 establishes a quasilinear
resonance result for Q (see [LaL] and [Wi]) at every L-pseudo-eigenvalue, a new concept.
The second direction of improvement over [Shi] occurs in Theorems 10-12 where at the first eigenvalue, super linear extensions of the main results of [Shi] are obtained via Theorem 9, which constitutes a new Sobolev-type compact imbedding for weighted spaces.
The third direction of improvement is in permitting singularities similar to those which arise in the study of higher order Bessel and associated Legendre functions, neither of which are allowed in [Shi]. These singularities are covered by the theorems of Chapter 1, and many examples using said functions are given.
Chapter 2 studies time-periodic singular quasilinear parabolic differential equations and extends results previously given in [LfS] and [LgS] and is motivated by semilinear results of the type in [BN] and [CL]. Theorem 1 obtains a double resonance time-periodic result for D^u-Qu at the higher eigenvalues of L without restriction where Q is an #H-perturbation of L. A number of examples are given to illustrate this new concept.
Theorem 2 of Chapter 2 deals with a time -periodic singular quasilinear reaction-diffusion system at the higher eigenvalues of the singular elliptic operator L. The idea of an H-L-pseudo-eigenvalue is introduced and two examples of quasilinear reaction-diffusion systems illustrating this concept are given, one of which is of activator-inhibitor type, an important notion in mathematical biology.