Very singular solutions for linear Dirichlet problems with singular convection terms

Abstract

We study the existence of distributional solutions for the boundary value problems (1.1) and (1.2) if E does not belong to LN, namely |E|≤|A||x|, A∈R. The size of A plays an important role: if α(N−2)≤|A|<α(N−1), we prove that if f∈L1(Ω) there exists a distributional solution u∈W01,q(Ω), for every q<Nα|A|+α<NN−1, of (1.1) (the case |A|<α(N−2) is studied in Boccardo (2015)). We then use this result to prove the existence of a bounded weak solution ψ of (1.2) if g(x)∈Lm(Ω), m>NαNα−|A|≥N2.