Abstract
The aim of this paper is to extend the usual framework of PDE with Au=-div ax,u,∇u to include a large class of cases with Au=∑β≤α-1βDβAβx,u,∇u,…,∇αu, whose coefficient Aβ satisfies conditions (including growth conditions) which guarantee the solvability of the problem Au=f. 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.
Highlights
This paper is motivated by the study of the unilateral problem associated with the following equation:A (u) + g (x, u) = f. (1)We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order 2m on a domain Ω in RN in generalized divergence form as follows:A (u) = ∑ (−1)|β|DβAβ (u, ∇u, . . . , ∇αu) . (2) |β|≤αThe function g satisfies a sign condition but has otherwise completely unrestricted growth with respect to u.Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type
We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order 2m on a domain Ω in RN in generalized divergence form as follows: A (u) = ∑ (−1)|β|DβAβ (u, ∇u, . . . , ∇αu)
Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation
Summary
This paper is motivated by the study of the unilateral problem associated with the following equation:. 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 α > 1. Brezis and Browder [5] used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution They showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). Benkirane and Gossez established this result in the Orlicz-Sobolev spaces WαLA(RN), see [8,9,10] It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent Wα,p(⋅)(RN) of the harder higher order case α > 1.
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