Very singular solutions for linear Dirichlet problems with singular convection terms

Gradient estimates for nonlinear elliptic equations with first order terms

“…Assumption (2.5), up to the addition of a whichever bounded vector filed, prescribes a threshold on the M N (Ω)-norm of E (see also [7] and [12]). As already said in the introduction, this smallness condition is sharp and cannot be weakened (see [17] and Remark 4.5). To clarify the different notation between (1.10) and (2.5), let us notice that…”

“…In the borderline case γ(B) = N −2m N m estimates with logarithmic corrections are obtained. This argument shows that if (2.5) is not satisfied the standard relation between the regularity of the data and the solution is lost (for a more detailed description of this fact see [17]). Now we are in the position of proving Theorems 2.1 and 2.2.…”

“…the standard relation between the regularity of f and u is lost and the regularity of the solution depends on the value of the Marcinkiewicz norm of E (see [17] and Remark 4.5). Let us also notice that the (1.10) is more general then (1.4).…”

Regularizing effect of the interplay between coefficients in Dirichlet problems with convection or drift terms

“…where 0 ∈ Ω, µ ∈ R + , A ∈ R. The inequality (1.10) where a(x) = µ u |x| 2 and E(x) = A x |x| 2 is satisfied by the real numbers R > 0 such that A 2 µ ≤ R. In [13], the existence of very singular distributional solutions u ∈ W 1,q 0 (Ω), for every q <…”

Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

“…Furthermore, in the same paper, the existence of entropy solutions to problem (5) also be considered provided E ∈ ðL 2 ðΩÞÞ N and f ∈ L 1 ðΩÞ. Recently, continuation of [2], Boccardo and Orsina [3] studied the existence of distributional solution u ∈ W 1,q 0 ðΩÞ to problem (5) with q < Nα/A + α provided αðN − 2Þ ≤ A < αðN − 1Þ and f ∈ L 1 ðΩÞ. Moreover, u verifies a prior estimation:…”