Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities - Mohammed El Mokhtar Ould El Mokhtar (Journal of Function Spaces)
This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities,  Δp2u=uγ−1u+μu−1−α/xβu in Ω and u=∂u/∂n=0 on ∂Ω, where 4<2p<N,0∈Ω, −∞<μ<μ∗,=N−2p1−α/pN,p<γ<p∗=pN/N−2p  is the critical Sobolev exponent,  0≤β<Nγ+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⊂ℝN be a bounded, smoothandstrictlystar-shapeddomain, β=0 and γ≥γ∗,=pN1−α/N−2p1−α−μNp>p∗=pN/N−2p.