We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.

Variants of Kato’s inequality are proved for general quasilinear elliptic operators L. As an outcome we show that, dealing with Liouville theorems for coercive equations of the type Lu = f (x, u,Du) on Ω ⊂ R^N , where f is such that f(x, t, ξ) t ≥ 0, the assumption that the possible solutions are nonnegative involves no loss of generality. Related consequences such as comparison principles and a priori bounds on solutions are also presented. An underlying structure throughout this work is the framework of Carnot groups.

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.

Let L be a general second order differential elliptic operator. By using a quasilinear version of Kato’s inequality, we prove that the only weak solution of the problem L(u) = |u|^(q−1) u on RN , q > p − 1, is u = 0. Here p ≥ 1 is related to L.

We prove comparison principles, uniqueness, regularity and symmetry results for p-regular distributional solutions of quasilinear very weak elliptic equations of coercive type and to related inequalities. The simplest model examples are -Δ_pu+|u|^(q-1)u=h on R^N, where q>p-1>0 and -div(\nabla u/\sqrt(1+|\nabla u|^2)+|u|^(q-1)u=h on ℝN, with q>0 and h∈L^1_loc(R^N).

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.

We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on RN. Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Henon-Lane-Emden system.

We prove a simple sufficient criteria to obtain some Hardy inequalities on Rie- mannian manifolds related to quasilinear second-order differential operator ∆p u := div | u|p−2 u . Namely, if ρ is a nonnegative weight such that −∆p ρ ≥ 0, then the Hardy inequality c M |u|p | ρ|p dvg ≤ ρp | u|p dvg , ∞ u ∈ C0 (M ). M holds. We show concrete examples specializing the function ρ. Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo- Nirenberg inequalities, uncertain principle and first order Caffarelli-Kohn-Nirenberg interpolation inequality.

Let u be a solution of the system of PDE L (u) = f(u) in R N , where L is a quasilinear second order elliptic operator in divergence form and f a given function. Our aim is to find uniform bounds for all possible solutions u of the system. In this paper we prove some bounds which are universal, in the sense that they are related only to the zeros of the nonlinearity f. Among others, the results apply to Allen– Cahn equation, Ginzburg–Landau systems, Gross–Pitaevskii systems and Lichnerowicz’s type equations.

In this paper we study the problem \[ \begin{cases}{\mathcal L}_\mu[u]:=\Delta^2u -\mu\frac{u}{|x|^4}=\lambda u +|u|^{2^*-2}u\quad\hbox{in\ }\Omega\\ u=\frac{\partial u}{\partial n}=0\quad\hbox{on\ }\partial\Omega\end{cases} \] where \Omega ⊂ R^n is a bounded open set containing the origin, n ≥ 5 and 2^∗ = 2n/(n − 4). We find that this problem is critical (in the sense of Pucci–Serrin and Grunau) depending on the value of μ ∈ [0, μ), μ being the best constant in Rellich inequality. To achieve our existence results it is crucial to study the behavior of the radial solutions (whose analytic expression is not known) of the limit problem Lμ u = u^(2* −1) in the whole space R^n . On the other hand, our non–existence results depend on a suitable Pohozaev-type identity, which in turn relies on some weighted Hardy–Rellich inequalities.

Let f : R → R be a continuous function. It is shown that under certain assumptions on f and A : R → R+ weak C 1 solutions of the differential inequality -div(A(|\nabla u|)\nabla u)\ge f(u) on RN are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot groups are considered.

In this paper, we prove the following result. Let α be any real number between 0 and 2. Assume that u is a solution of {(-δ)α/2u(x)=0,x∈Rn,lim¯|x|→∞u(x)|x|γ≤0, for some 0≥≥;1 and γα. Then u must be constant throughoutRn. This is a Liouville Theorem for α-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only α-harmonic functions are affine.

We study the uniqueness problem of σ-regular solution of the equation, −∆p u + |u|q−1 u = h on RN , where q > p − 1 > 0. and N > p. Other coercive type equations associated to more general differential operators are also investigated. Our uniqueness results hold for equations associated to the mean curvature type operators as well as for more general quasilinear subelliptic operators.

We study the uniqueness problem of the equation, -\Delta L;p u C jujq 1 u D h on RN ; where q > p 1 > 0: and N > p: Uniqueness results proved in this paper hold for equations associated to the mean curvature type operators as well as for more general quasilinear coercive subelliptic problems