Positive solutions for Robin problem involving the p(x)-Laplacian☆

By Shao-Gao Deng. 2009

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Abstract Consider Robin problem involving the p ( x ) -Laplacian on a smooth bounded domain Ω as follows { − Δ p ( x ) u = λ f ( x , u ) in Ω , | ∇ u | p ( x ) − 2 ∂ u ∂ η + β | u | p ( x ) − 2 u = 0 on ∂ Ω . Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ * > 0 such that the problem has at least two positive solutions if λ ∈ ( 0 , λ * ) , has at least one positive solution if λ = λ * + ∞ and has no positive solution if λ > λ * . To prove the results, we prove a norm on W 1 , p ( x ) ( Ω ) without the part of | ⋅ | L p ( x ) ( Ω ) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7f65c19f8a53a939c9ea19466dc3eaecf72fcb1