Existence and nonexistence of positive radial solutions of superlinear elliptic systems

Abstract

The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic systemwhere Ω is a ball in R N and f , g are positive continuous functions satisfying f (x, 0, 0) = g(x, 0, 0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use the topological degree theory combined with the blow up method of Gidas and Spruck. When Ω = R N , we give some sufficient conditions of nonexistence of radial positive solutions for Liouville systems.

On positive solutions for a class of quasilinear elliptic systems

“…Assuming the above conditions we collect some properties of solutions of S ϕ which will be needed for our later analysis. 1], and (z, u) is a solution of (S ϕ ). Then the following properties hold:…”

“…We can mention, among others, [1], [2], [9] where nonlinearities satisfy special growth or [11] in which the authors describe necessary conditions for the existence of bounded positive solutions for the system with Laplace operator. This case is also discussed by J. M. doÓ et al (in [3], [4], [5], [6]), who apply topological methods like fixed point theorem due to Krasnosielskii, fixed point index theory and upper-lower solution methods.…”

Nonlinear BVPS with functional parameters

“…Recently the research on positive (nonnegative) solutions of the systems of nonlinear equations containing Laplacian or perturbed p-Laplace operator has been very active and enjoying of increasing interest (see e.g. [1,2,11,13,[17][18][19] and references therein).…”

“…[5,14]), where the generalized eigenvalue problem is discussed. We want to present the methods which allow us to study the systems of elliptic equations in both annular domain (as in (1) given below) and exterior domain (as in (2) given below). Moreover due to the fact that we do not need any information concerning behavior of nonlinearity at infinity, we cover both suband superlinear problems.…”

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities

“…Semilinear scalar elliptic equations with concave and convex nonlinearities are widely studied; we refer the readers to [2,3,7,13,14,21] etc.. For the semilinear elliptic systems, we refer to Ahammou [1,5,6,9,12,15,17,19]. The model type is written as follows: (2) When h ≡ g ≡ 1, Eq.…”