A remark on entire solutions of quasilinear elliptic equations

Abstract

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below. Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].

Entire solutions of completely coercive quasilinear elliptic equations, II

“…A result similar to Theorem 10 was proved in [9] for a more special class of operators A, but with a weaker growth condition than (17), namely u(x) = o(|x| κ ) as |x| → ∞, see Theorem 1.1 of [9].…”

Nonexistence of nonnegative solutions of elliptic systems of divergence type

“…Recent generalizations of this result to a broad class of weakly elliptic operators, including as special cases the mean curvature and p-Laplace operators, is due to Farina and Serrin in [10] and D'Ambrosio and Mitidieri [7,8]. For a related nonexistence result we mention the recent paper of Serrin [26].…”

“…Sinceũ → u τ andṽ → v τ in L 1 loc (R N ) as → 0, then by the Lebesgue dominated convergence theorem we arrive to (25) and (26), respectively.…”

“…Finally, this approach, via the main comparison theorem of [24], allowed Pucci and Serrin in [25] to cover several of the important cases left open in [10].…”

On the role of gradient terms in coercive quasilinear differential inequalities on Carnot groups

“…Our approach to (P1) has its roots in the works [31,30,32,26] by the third author and his collaborators, and in the subsequent improvements in [27,21]. Interesting Liouville theorems for slowly growing solutions have also been shown in [12,28,9] for a broad class of differential inequalities including (1.12). However, as we shall see, the results in [12,28,9] are skew with our Theorem 1.4, that is, the range of parameters considered is quite different from our's.…”