The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

We prove the existence of extremal solutions of the following quasilinear elliptic problem −∑i=1(∂/∂xi)ai(x, u(x), Du(x)) + g(x, u(x), Du(x)) = 0 under Dirichlet boundary condition in Orlicz-Sobolev spacesW1 0 LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g : Ω ×R ×RN 󳨀→ R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.


Introduction
The aim of this paper is to study some qualitative properties of solutions of the following quasilinear elliptic problem: (,  () ,  ()) +  (,  () ,  ()) = 0 in on a bounded domain Ω ⊂ R  with a Lipschitz boundary Ω in Orlicz-Sobolev spaces.The differential part is driven by a Leray-Lions operator, while the nonlinear term  : Ω × R × R  → R is a Carathéodory function satisfying a growth condition.
Hence, the growth condition (3) is not more general than (2).When trying to weaken the restriction on the Leray-Lions operator and the growth condition (2), one is led to replace  1, 0 (Ω) with  1 0   (Ω) built from an Orlicz space   (Ω) instead of   (Ω), where  is an -function.The choice [2,3]).Many papers used the surjectivity result for pseudomonotone operators (see, e.g., [1,Theorem 2.99]) defined on reflexive spaces to prove the existence of the solution (see, e.g., [1,2,4,5]).Our method does not need the reflexivity of the spaces.It is well known that the Orlicz space is reflex if and only if both  and its complementary function  satisfy Δ 2 -condition.However, there exist many spaces without reflexivity.For example, let In this paper, we get rid of the restriction of the reflexivity of the spaces and get a weak solution for (1) in Orlicz-Sobolev spaces by using a linear functional analysis method.We also give the enclosure of solutions and prove the existence of extremal solutions.
This paper is organized as follows.Section 2 contains some preliminaries and some technical lemmas which will be needed.In Section 3, we use the linear functional analysis method to prove the existence of solutions for (1) in separable Orlicz-Sobolev spaces and the sub-supersolutions method to give the enclosure of solutions and the existence of extremal solutions between a subsolution and a supersolution.We also get the compactness and directness of the solutions set.
,  are called the right-hand derivatives of , , respectively.
We will extend these -functions into even functions on all R.

Orlicz Spaces.
Let Ω be an open and bounded subset of R  and  be an -function.The Orlicz class K  (Ω) (resp., the Orlicz space   (Ω)) is defined as the set of (equivalence classes of) real valued measurable functions  on Ω such that   () < +∞(resp.  (/) < +∞ for some  > 0).  (Ω) is a Banach space under the (Luxemburg) norm: and The dual space of   (Ω) can be identified with   (Ω) by means of the pairing ∫ Ω ()V(), and the dual norm of   (Ω) is equivalent to ‖ ⋅ ‖ () .
where diam Ω is the diameter of Ω.
Consider the following nonlinear elliptic equation: Here,  : Ω × R × R  → R is assumed to be a Carathéodory function.