Abstract
The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|β|^p β2^β) + Ξ»k~i~ (|x |) f^i^ (u~1~, β¦,u~n~) = 0, p > 1, R~1~ < |x | < R~2~, u~i~ (x) = 0, on |x | = R~1~ and R~2~, i = 1, β¦, n, x β β^N^ , where k~i~ and f^i^, i = 1, β¦, n, are continuous and nonnegative functions. Let u = (u~1~, β¦, u~n~), Ο (t) = |t |^p β2^t, f^i^~0~ = lim~βuββ0~((f^i^ (u))/(Ο (βuβ))), f^i^~β~= lim~βuβββ~((f^i^ (u))/(Ο (βuβ))), i = 1, β¦, n, f = (f^1^, β¦, f^n^), f~0~ = β^n^ ~i =1~ f^i^ ~0~ and f~β~ = β^n^ ~i =1~ f^i^ ~β~. We prove that either f~0~ = 0 and f~β~ = β (superlinear), or f~0~ = βand f~β~ = 0 (sublinear), guarantee existence for all Ξ» > 0. In addition, if f^i^ (u) > 0 for βuβ > 0, i = 1, β¦, n, then either f~0~ = f~β~ = 0, or f~0~ = f~β~ = β, guarantee multiplicity for sufficiently large, or small Ξ», respectively. On the other hand, either f~0~ and f~β~ > 0, or f~0~ and f~β~ < β imply nonexistence for sufficiently large, or small Ξ», respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (Β© 2007 WILEYβVCH Verlag GmbH & Co. KGaA, Weinheim)