Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

We study the quasilinear elliptic inequality \[ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, \] where $p>0$ , $q, \mu \in \mathbb {R}$ , $m>1$ and $I_\alpha$ is the Riesz potential of order $\alpha \in (0,N)$ . We obtain necessary and sufficient conditions for the existence of positive solutions.

On Absence of Nonnegative Monotone Solutions for Some Coercive Inequalities in a Half-Space

Using the nonlinear capacity method, we investigate the problem of absence of nonnegative monotone solutions for a quasilinear elliptic inequality of type Δpu≥uq in a half-space in terms of parameters p and q.

Singular semilinear elliptic inequalities in the exterior of a compact set

We study the semilinear elliptic inequality –Δu ≥ φ(δK (x))f(u) in ℝN / K, where φ, f are positive and non-increasing continuous functions. Here K ⊂ ℝN (N ≥ 3) is a compact set with finitely many components, each of which is either the closure of a C2 domain or an isolated point, and δK (x) = dist(x, ∂K). We obtain optimal conditions in terms of φ and f for the existence of C2-positive solutions. Under these conditions we prove the existence of a minimal solution and we investigate its behaviour around ∂K as well as the removability of the (possible) isolated singularities.

Nonexistence Results of Positive Entire Solutions for Quasilinear Elliptic Inequalities

This paper treats the quasilinear elliptic inequalitywhere N ≥ 2, m > 1, σ >m− 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.