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Positive solutions and eigenvalue intervals for nonlinear systems

Authors:

Affiliation:

(1) Department of Mathematical Sciences, Tsinghua University, Beijing, 100 084, China;(2) Department of Applied Mathematics, Hohai University, Nanjing, 210 098, China;(3) Department of Mathematics, National University of Ireland, Galway, Ireland

Abstract:

This paper deals with the existence of positive solutions for the nonlinear system

$$
(q(t)phi (p(t)u'_i (t)))' + f^i (t,u) = 0, 0 < t < 1, i = 1,2, ldots ,n
$$

. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here u = (u ₁, …, u _n) and f ⁱ, i = 1, 2, …, n are continuous and nonnegative functions, p(t), q(t): [0, 1] → (0, ∞) are continuous functions. Moreover, we characterize the eigenvalue intervals for

$$
(q(t)phi (p(t)u'_i (t)))' + lambda h_i (t)g^i (u) = 0, 0 < t < 1, i = 1,2, ldots ,n
$$

. The proof is based on a well-known fixed point theorem in cones.

Keywords:

Nonlinear system p-Laplacian positive solutions eigenvalue intervals fixed point theorem in cones

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