Singular dimension of the solution set of a class of p-Laplace equations

摘要

We consider the boundary value problem -Δ_pu=F(x), u∈W~(1,p)0 (Ω) in a bounded open domain Ω r{double-struck}~N, where F∈L~(P')(Ω), 1<∞, p' =p/(p-1). Let X(Ω, p) be the set of weak solutions u generated by all right-hand sides F. Define the singular dimension of the solution set as the supremum of Hausdorff dimension of singular sets of solutions in X(Ω, p), and denote it by s-dim X(Ω, p). We show that for p<2 we have s-dim X(Ω, p)=(N-pp')~+, where r~+=max{0, r}. In the proof we exploit among others a regularity result for p-Laplace equations due to J. Simon [Sur des équations aux Dérivées Partielles Non Linéaires, Thése, Paris, 1977], involving Besov spaces. For 1<2, an estimate for the singular dimension of the solution set is obtained.