Existence Results for a Perturbed Dirichlet Problem Without Sign Condition in Orlicz Spaces
- PDF / 233,670 Bytes
- 22 Pages / 594 x 792 pts Page_size
- 65 Downloads / 142 Views
EXISTENCE RESULTS FOR A PERTURBED DIRICHLET PROBLEM WITHOUT SIGN CONDITION IN ORLICZ SPACES H. Moussa,1,2 M. Rhoudaf,3 and H. Sabiki 4
UDC 517.5
We deal with the existence result for nonlinear elliptic equations of the form Au + g(x, u, ru) = f, where the term −div a(x, u, ru) is a Leray–Lions operator from a subset of W01 LM (⌦) into its dual. The growth and coercivity conditions on the monotone vector field a are prescribed by an N -function M, which not necessarily satisfies a ∆2 -condition. Therefore, we use Orlicz–Sobolev spaces that are not necessarily reflexive and assume that the nonlinearity g(x, u, ru) is a Carath´eodory function satisfying solely a growth condition without any sign condition. The right-hand side f belongs to W −1 EM (⌦).
1. Introduction In the last decade, there has been an increasing interest in the study of various mathematical problems in modular spaces. These problems have numerous applications [13, 33, 34]. Actually, they resulted in a renewal of the interest in Modular spaces whose origins can be traced back to the work of Orlicz published in the 1930s. In the 1950s, the investigations in this field were carried out by Nakano [29]. Later, Polish and Czechoslovak mathematicians studied the modular function spaces (see, e.g., [25, 28]). One of our motivations to study nonlinear problems in modular spaces comes from the applications to electrorheological fluids as an important class of non-Newtonian fluids (sometimes referred to as smart fluids). The electrorheological fluids are characterized by the ability to undergo significant changes in their mechanical properties under the influence of external electromagnetic fields. A mathematical model of these fluids was proposed by Rajagopal and R˙uzˇ iˇcka [32, 33]. We refer the reader, e.g., to [4, 9, 21–24, 27] for different nonstandard growth conditions. Another important application is related to image processing [30], where the same kind of diffusion operator is used to underline the borders of the distorted image and to eliminate the noise. In the present paper, we are interested in proving existence results for a nonlinear elliptic problem with nonlinearity. Our investigations are carried out for the case of fairly general growth conditions for the highest-order term. This formulation requires a general framework for the function space setting. The problems are considered in the Orlicz spaces. The level of generality of our considerations has a crucial significance for the applied methods. This is a natural generalization of numerous recent studies carried out in the Lebesgue and Sobolev spaces, which can be regarded as a particular case of our approach. Let ⌦ be a bounded open set in RN , N ≥ 2. We consider the following nonlinear elliptic problem: Au + g(x, u, ru) = f u = 0 on
in ⌦, (1.1)
@⌦,
1
University of Sultan Moulay Slimane, Beni-Mellal, Morocco; e-mail: [email protected]. Corresponding author. 3 University of Moulay Ismail, Meknes, Morocco; e-mail: [email protected]. 4 University of Ibn Tofail, K´en
Data Loading...
