Degenerate p(x)p(x)-elliptic equation with second membre in L^1

Abstract

In this paper, we prove the existence of a solution of the strongly nonlinear degenerate p(x)-elliptic equation of type:where Ω is a bounded open subset of IR N , N ≥ 2, a is a Carathéodory function from Ω × IR × IR N into IR N , who satisfies assumptions of growth, ellipticity and strict monotonicity. The nonlinear term g: Ω × IR × IR N −→ IR checks assumptions of growth, sign condition and coercivity condition, while the right hand side f belongs to L 1 (Ω) .

On positive weak solutions for a class of weighted (p(.), q(.))−Laplacian systems

“…− div (ω(x)|∇v| q(x)−2 ∇v) = 1 in Ω\Ω δ . (3.17) On the other hand, we have 18) and 20) and 1) . (3.21) Thus, according to (1.3) and (1.4), we obtain…”

“…1) where d(x) = dist(x, ∂Ω) and η : Ω −→ R is a continuous function such that lim and p, q ∈ C(Ω) with 1 < p − p + < N and 1 < q − q + < N, A weak solution of (P 1 ) is a pair (u, v) ∈ W…”

Anisotropic Elliptic Nonlinear Obstacle Problem with Weighted Variable Exponent

“…(Ω, − → w (.)). By an adapted method of that of Adams [1], and by constructing an isometric isomorphism from…”

“…The study of (P ) is a new and interesting topic when the data is in L 1 . One result on this topic can be found in [5,8,11], where the discussion was conducted in the framework of weighted anisotropic Sobolev space with variable exponent (we refer to [1,2,11] for more details), the notion of a entropy solution was introduced by Benilan et. al [7,9] and P.-L. Lions [14] in their study of the Boltzmann equation.…”

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

“…By combining variational methods based on critical point theory, with suitable truncation techniques and flow invariance arguments. Some important and interesting results can be found in, for example [1,3,14,16,20,27]. Moreover, S. Liu [25] by using Morse theory, have established the existence of weak solutions in the case a(x, u, ∇u) = |∇u| p−2 u and H(x, u, ∇u) = f (x, ∇u) with Dirichlet boundary conditions.…”