Positive radial solutions for p-Laplacian systems

Summary.

The paper deals with the existence of positive radial solutions for the p-Laplacian system div(|∇ u _i |^p-2∇ u _i )  +  f ⁱ(u ₁, ..., u _n )  =  0, |x|  <  1, u _i (x)  =  0, on $|x| = 1, i = 1, \ldots, n, p > 1, x \in {{\mathbb{R}}}^N$ . Here f ⁱ, i  = 1,...,n, are continuous and nonnegative functions. Let ${\bf u}= (u_1, \ldots, u_n), \| {\bf u}\| = \sum^n_{i=1}|u_i|, f^i_0 = \lim_{\| u \|\rightarrow 0} \frac{f^i(u)}{\| {\bf u}\| ^{p-1}}, f^i_\infty = \lim_{\| {\bf u}\| \rightarrow \infty} \frac{f^i(u)}{{\|\bf u}\| ^{p-1}}, i = 1, \ldots , n, {\bf f} = (f^1, \ldots , f^n), {\bf f}_0 = \sum^n_{i=1} f^i_0$ and f _∞  =  ∑ⁿ _i=1 f ⁱ _∞. We prove that f ₀  = ∞ and f _∞ = 0 (sublinear), guarantee the existence of positive radial solutions for the problem. Our methods employ fixed point theorems in a cone.