The fractional p(., .)-Neumann boundary conditions for the nonlocal p(., .)-Laplacian operator. Issue 3 (11th February 2023) - Irzi, Nawal; Kefi, Khaled
Abstract : In this paper, we deal with the fractional p ( ., . ) -Laplacian problem with nonlocal Neumann boundary conditions 1 { ( − Δ ) p ( ., . ) s u + | u | p ¯ ( x ) − 2 u = λ V 1 ( x ) | u | q ( x ) − 2 u, in Ω, N s, p ( ., . ) u = μ V 2 ( x ) | u | r ( x ) − 2 u, on R N ∖ Ω, here λ, μ > 0 are parameters and V 1, V 2 are two nonnegative weighted functions. The domain Ω ⊂ R N ( N ≥ 1 ) is smooth and bounded, p ¯ ( x ) = p ( x, x ), q, r are continuous bounded functions. Applying the Ekeland principle and the variational method, under appropriate assumptions, we show that the above problem has nontrivial weak solutions.