QUASILINEAR ELLIPTIC EQUATIONS AND INEQUALITIES WITH RAPIDLY GROWING COEFFICIENTS

Abstract

Let us consider the boundary value problem

formula here

where Ω ⊂ ℝ^N is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let

formula here

Then, if ϕ(s) = [mid ]s[mid ]^p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W^1,₀^p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that

formula here

and

formula here

then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].

It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F[ratio ]Ω×ℝ→ℝ is a Carathéodory function that satisfies certain growth conditions to be specified later.

We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).

We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.