Semilinear Elliptic Systems with Dependence on the Gradient

Abstract

We provide results on the existence, non-existence, multiplicity and localization of positive radial solutions for semilinear elliptic systems with Dirichlet or Robin boundary conditions on an annulus. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.2010 Mathematics Subject Classification. Primary 35B07, secondary 35J57, 47H10, 34B18.

Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria

“…In this paper we establish new criteria for the existence of positive radial solutions for the system of BVPs where Ω = {x ∈ R n : R 0 < |x| < R 1 } is an annulus, 0 < R 0 < R 1 < +∞, the nonlinearities f i are non-negative continuous functions and ∂ ∂r denotes (as in [12]) differentiation in the radial direction r = |x|. The problem of the existence of positive radial solutions of elliptic equations having nonlinearities that depend on the gradient, subject to Dirichlet or mixed boundary conditions, has been investigated, via different methods, by a number of authors, for example in [4,5,6,7,8,11,31]. We seek radial solutions of the system (1.1) by means of an auxiliary system of nonlinear Hammerstein integral equations using the fixed point index theory and the invariance properties of the involved cone.…”

“…In the recent paper [6] in collaboration with Pietramala, we worked in the Banach space 1], where the weights ω i are suitable nonnegative and continuous functions on [0, 1]. In the special case ω(t) = t(1 − t), the space C 1 ω [0, 1] is utilized by Agarwal and others in [1].…”

“…The problem of the existence of positive radial solutions of elliptic equations having nonlinearities that depend on the gradient, subject to Dirichlet or mixed boundary conditions, has been investigated, via different methods, by a number of authors, for example in [4,5,6,7,8,11,31]. We seek radial solutions of the system (1.1) by means of an auxiliary system of nonlinear Hammerstein integral equations using the fixed point index theory and the invariance properties of the involved cone.…”

“…Theorem 3.1 [6]. Suppose that, for i = 1, 2, there exist ρ i , s i ∈ (0, +∞), with ρ i < c i s i , such that the following conditions hold sup…”

Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains

“…Cianciaruso and co-authors [8,9,12], De Figueiredo and co-authors [13,14] and Faria and co-authors [22].…”