Entire solutions of quasilinear elliptic systems on Carnot groups

Abstract

We prove general a priori estimates of solutions of a class of quasilinear elliptic system on Carnot groups. As a consequence, we obtain several non-existence theorems. The results are new even in the Euclidean setting.

On some multicomponent quasilinear elliptic systems

“…The results proved in this paper are a substantial generalization of those proved in [4] for the scalar case, and [5] and [6] for systems of two equations. However, to keep our exposition simple and transparent we bound our interest to systems with three equations.…”

“…In this paper we prove some Liouville theorems of general quasilinear second order elliptic systems and inequalities in R N in divergence form. The interested reader may refer to [11], [10] for earlier results related to this work and to [5] and [6] for more recent outcomes.…”

Some recent results on singular $ p $-Laplacian systems

“…Unfortunately, bounds from above alone do not allow to get a solution of (32): treating singular terms additionally requires some estimates from below. Theorem 3.1 in [19] ensures that solutions to (34) turn out locally greater than a positive constant regardless of ε. Thus, under the hypotheses below, one can construct a sequence {(u ε , v ε )} ⊆ X p,q (R N ) such that (u ε , v ε ) solves (34) for all ε > 0 and whose weak limit as ε → 0 + is a distributional solution to (32).…”

On properties of solutions of quasilinear second-order elliptic inequalities

“…Proof of Lemmas 3.2 and 3.3 can be found in [12,13,14] and [1], respectively. Lemma 3.4 is given in [2, Lemma 3.1].…”

“…The absence of nontrivial global solutions of differential equations and inequalities or, in other words, the blow-up phenomenon, traditionally attracts the attention of mathematicians [1][2][3][4][5][6][7][8][9][10][11]. We obtain exact conditions on the function f guaranteeing that any non-negative solution of (1.1), (1.3) is identically zero.…”