Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities

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 .