Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities. (13th March 2023) - Ould El Mokhtar, Mohammed El Mokhtar
Abstract : This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities,   Δ p 2 u = u γ − 1 u + μ u − 1 − α / x β u   in   Ω and u = ∂ u / ∂ n = 0   on   ∂ Ω, where 4 < 2 p < N, 0 ∈ Ω,   − ∞ < μ < μ ∗, = N − 2 p 1 − α / p N, p < γ < p ∗ = p N / N − 2 p   is the critical Sobolev exponent,   0 ≤ β < N γ + α / γ + 1,   0 < α < 1 . Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω ⊂ ℝ N be a bounded, smoothandstrictlystar-shapeddomain, β = 0   and   γ ≥ γ ∗, = pN 1 − α / N − 2 p 1 − α − μ Np > p ∗ = pN / N − 2 p .