Integrability of Solutions of Nonlinear Elliptic Equations with Right-Hand Sides from Logarithmic Classes

On Conditions of the Nonexistence of Solutions of Nonlinear Equations with Data from Classes Close to <em>L</em>¹

“…In this connection we observe that actually the conclusion of Theorem 3.1 holds if in the conditions of the theorem we assume that the inequality p 2−1/n is satisfied instead of the inequality p > 2−1/n (see [ This result was established by the first author in [2, Theorem 1.5.6]. The same conclusion as in the given theorem under the conditions p 2−1/n and n/(np−n+1) < m < p * /(p * −1) has already been obtained in [9].…”

On the convergence of functions from a Sobolev space satisfying special integral estimates

“…An equality of the form (4) is used in [1] to determine the gradient of elements of a functional class containing the set • T 1,p (Ω). The direct definition of the functions δ i u for u ∈ • T 1,p (Ω) by equality (3) and the proof of Proposition 1 can be found in [2]. Lemma 1 is, in fact, proved in [3].…”

“…Lemma 1 is, in fact, proved in [3]. For similar results with more exact estimates used instead of (5), see [4,5]. For θ = 1, inequalities (7) and (8) are established under condition (6) in [1].…”

Regularity results for the gradient of solutions linear elliptic equations with L 1, λ data

“…Finally, we remark that L 1,λ ( ) is not contained nor contains L log L( ) as well as L 1 n n−1 ( ) (see Remark 2.3 and Appendix); consequently, our regularity result is independent of that studied in [5,6,11,17].…”

“…We remark that a solution u of the problem (1) (very weak or entropy or distributional) belongs only to 1≤q< n n−1 W 1,q 0 ( ); the best regularity for Du, that is Du ∈ L n n−1 ( ), can be achieved under the additional assumption f ∈ L log L( ) or f ∈ L 1 n n−1 ( ) (see [5,6,11,17,21,22]). …”