Weak and entropy solutions for a class of nonlinear inhomogeneous Neumann boundary value problem with variable exponent
- Stanislas Ouaro souaro@univ-ouaga.bf
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https://doi.org/10.4067/S0719-06462012000200002Abstract
We study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous Neumann boundary value problem involving the ð‘(ð‘¥)-Laplace of the form − div É‘(ð‘¥, ∇ð‘¢) + |ð‘¢| ð‘(ð‘¥)−2 ð‘¢ = f in Ω, É‘(ð‘¥, ∇ð‘¢).η = 𜑠on ∂Ω, where Ω is a smooth bounded open domain in â„N, N ≥ 3, ð‘ ∈ C(Ω) and ð‘(ð‘¥) > 1 for 𑥠∈ Ω. We prove the existence and uniqueness of a weak solution for data 𜑠∈ L(ð‘−) ”² (∂Ω) and f ∈ L(ð‘−) ”² (Ω), the existence and uniqueness of an entropy solution for L1−data f and 𜑠independent of ð‘¢ and the existence of weak solutions for f dependent on ð‘¢ and 𜑠∈ L(ð‘−) ”² (Ω).
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