| Contents |
| We study the existence of a positive radial solution to the nonlinear eigenvalue problem:
$$\begin{array}{lcl}
-\Delta u(x) & = & \lambda k_1\left(\left|x\right|\right) f\left(v(x)\right); \; x \in \Omega_e\\
-\Delta v(x) & = & \lambda k_2\left(\left|x\right|\right)g\left(u(x)\right); \; x \in \Omega_e\\
\end{array}$$
$$\begin{array}{lcl}
u = 0 = v & ; & \left|x\right| = r_0 (> 0)\\
u \rightarrow 0, v \rightarrow 0 & ; & \left|x\right|\rightarrow \infty\\
\end{array}$$
where $\lambda > 0$ is a parameter, $\Delta u = \text{div}(\nabla u)$ is the Laplacian operator, $\Omega_e = \left\{ x \in {\mathbb R}^N \big| \left|x\right| > r_0, N > 2\right\}$ and $k_i \in C^1\left(\left[r_0, \infty\right), \left(0, \infty\right)\right)$: $i = 1,2$ are such that $k_i\left(\left|x\right|\right) \rightarrow 0$ as $\left|x\right| \rightarrow \infty$. Here $f:\left[0,\infty\right) \rightarrow {\mathbb R}$ and $g:\left[0,\infty\right) \rightarrow {\mathbb R}$ are $C^1$ nondecreasing functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for $\lambda \approx 0$ via degree theory and rescaling arguments. |
|