NONLINEAR WEIGHTED p-LAPLACIAN ELLIPTIC INEQUALITIES WITH GRADIENT TERMS

Abstract

In this paper, we give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div {g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)ℓ(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria of independent interest stated in the Introduction.

Keller–Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

“…We mention that, for the particular case of the p-Laplace operator, this result has also appeared as the first part of Theorem 1.5 of [14], where the authors deal with a weighted version of (19). As the reader can easily check, if the weights are trivial, then the two theorems coincide; we observe, however, that the C -monotonicity of l in (L) is a slightly milder requirement than the one in [14].…”

“…(13) In the last few years some authors (see, for example, [15,12,9,4]) have studied a generalization of the (14) which can be considered as a natural p-Laplace operator in the setting of the Heisenberg group.…”

Nonexistence of nonnegative solutions of elliptic systems of divergence type

“…Furthermore, if σ 1 = σ 2 = γ 1 = γ 2 = θ 1 = θ 2 = 0, then (15) reduces to the well known subcritical and critical condition p 1 N(p − 1)/(N − p) due to Mitidieri and Pohozaev for the general nonradial case in [21]. See [21] for further comments on previous results.…”

“…Furthermore, nonnexistence of nonnegative solutions of coercive inequalities of the type of (16) has been studied in the recent paper [15], where, among other facts including existence theorems, it is covered also the case, when a general monotone function (|Du|) on the right-hand side replaces the term |Du| θ , θ 0. In [15], as well as [13] and [14], the idea of the proof is completely different.…”

“…In this section we shall give the proof of the main theorem of the paper which concerns the so called superhomogeneous subcritical and critical case, indeed condition (15), under (H2), implies…”