Nonlinear Elliptic Systems with Coupled Gradient Terms

Abstract

In this paper, we analyze the existence and non-existence of nonnegative solutions to a class of nonlinear elliptic systems of type :where Ω is a bounded domain of IR N and p, q ≥ 1. f, g are nonnegative measurable functions with additional hypotheses and λ, µ ≥ 0. This extends previous similar results obtained in the case where the right-hand sides are potential and gradient terms, see [2,4].

On Some Nonlocal Elliptic Systems with Gradient Source Terms

“…The main goal of the present work is to study the existence and nonexistence of (weak) solutions to System (S). It is an extension of the results obtained in [10] for the local elliptic System (S) that is when s = 1, and those obtained in [5] pertaining to single nonlocal Hamilton-Jacobi equation (1.2). It is needless to say that this extension is far less trivial than it looks at first sight.…”

“…Proof. We closely follow the approach developed in [10,2,5] while taking into account the difficulties caused by the loss of regularity near the boundary of the domain in the nonlocal case.…”

“…In this paper, we look for natural conditions on (p, q), (f, g) and (λ, µ) which allow us to establish existence of solutions to System (S) as is done in [10] where the fourth author and his coworkers studied the local case (i.e. s = 1).…”

“…-First, in [10] the fourth author and his coworkers investigated the local version (s = 1) of System (S). The existence of nonnegative solutions is proved for any (p, q) ∈ [1, ∞) 2 such that pq > 1, under suitable integrability conditions of f and g, and smallness conditions on λ and µ.…”

A generalization of a criterion for the existence of solutions to semilinear elliptic equations

“…One can find in the literature several contributions about existence questions for equations of type (1.24) with p, q > 1 arbitrary and b = 0 or also b > 0, or for various extended versions. We refer for instance to [6], [2], [13], [12], [15], [3], [5] and to the numerous references in them.…”