In the remaining of this section, we will consider the almost periodicity and pseudo almost periodicity of the bounded weak solution of problem (4.1) under the assumption that f and h have the corresponding properties.
Proposition 3.3 Suppose that u ( x ) ∈ W loc 1, p is a bounded weak solution of (1.1), then osc B R 5 u ≤ γ osc B R u + C R 1 − n q (3.15).
Theorem 3.4 Suppose that u ( x ) ∈ W loc 1, p is a bounded weak solution of (1.1), then osc B r u ≤ 5 κ ( r R ) κ osc B R u + C r κ (3.16).
Theorem 3.9 Suppose that u ( x ) ∈ W loc 1, p is a bounded weak solution of (1.1), then osc Ω r u ≤ 5 κ ( r R ) κ osc Ω R u + C r κ (3.34).
Proposition 3.8 Suppose that u ( x ) ∈ W loc 1, p is a bounded weak solution of (1.1), then osc Ω R 5 u ≤ γ osc Ω R u + C R ϵ 1, where γ ∈ ( 0, 1 ) and C are positive constants depending on n, q, s and ϵ in (3.33), holds.
Theorem 1.1 Consider a bounded domain Ω in R N, with a boundary of class C 1. Let u be a bounded weak solution of (1.1) satisfying the boundary condition (H5), where the structure conditions (H1 - H3) hold for A i j α β, and (H4) holds for B i. Consider a fixed γ ∈ ( 0, σ ]. Then there exist positive R 1 and ε 0 (depending only on n, N, λ, L, b, M, a ( M ), ω and γ) with the property that.
For the existence of bounded weak solutions of equations (1.1) and (1.2), we refer to the work of Fournier et al. [14].
We get a theorem which shows the existence of a bounded weak solution for this problem.
In the case (alpha(0)=0), (fin W^{-1,r}(Omega cap L^{1}(Omega)) with (rgeq p'), (r>frac{N}{p-1} ), Rakotoson proved the existence of a bounded weak solution to problem ((mathscr{P})) (see [12]), provided that F satisfies a sign condition.
We also remark that the existence of bounded weak solutions to this type of problems have been studied in [9 11], assuming that (p(x in C bar{Omega})).
As (fin L^{q}(Omega)) with (qgeqmax{1,frac{N}{p}}), we shall give a direct method to prove the existence of bounded weak solutions to problem ((mathscr{P})) in the standard sense, i.e. (uin W_{0}^{1,p}(Omega)).
