The aim of this paper is to extend the usual framework of PDE with to include a large class of cases with , whose coefficient satisfies conditions (including growth conditions) which guarantee the solvability of the problem . This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation. 1. Introduction This paper is motivated by the study of the unilateral problem associated with the following equation: We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order on a domain in in generalized divergence form as follows: The function satisfies a sign condition but has otherwise completely unrestricted growth with respect to . Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type. For , that is, of second order, their solvability under fairly general and natural assumptions was proved by Hess [2]. The treatment of the case is more involved due to the lack of a simple truncation operator in higher order Sobolev spaces. Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation. This yielded the solvability of (1) for . Brezis and Browder [5] then used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution. They also showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). In a more general case, Boccardo et al. studied inequations associated with (1), see [7]. The functional setting in all the results mentioned above is that of the usual Sobolev spaces , and the functions in (2) are supposed to satisfy polynomial growth conditions with respect to and its derivatives. Benkirane and Gossez established this result in the Orlicz-Sobolev spaces , see [8–10]. It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent of the harder higher order case . We consider problem (1) as well as Hedberg’s approximation theorem and Brezis-Browder’s question on the action of some distributions. The paper is structured as follows. After some necessary preliminaries, in Section 3, we give the proof of the
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