On a class of nonvariational elliptic systems with nonhomogenous boundary conditions

Abstract

Using a fixed--point theorem of cone expansion/compression type, we show the existence of at least three positive radial solutions for the class of quasi-- linear elliptic systems \begin{equation*} \left\{ \begin{array}{rclcl} -\Delta_p u & = & \lambda k_1(|x|) f(u,v) & \mbox{ in } & \Omega, \\ -\Delta_q v & = & \lambda k_2(|x|) g(u,v) & \mbox{ in } & \Omega, \\ ( u,v) & = & (a,b) & \mbox{ on } & \partial \Omega \end{array} \right. \end{equation*} where the nonlinearities $ f, g \in C([0, +\infty)^2; [0,+\infty)) $ are superlinear at zero and sublinear at $ + \infty. $ The parameters $\lambda, a $ and $ b $ are positive, $ \Omega $ is the ball in $\mathbb{R}^N$, with $ N \ge 3, $ of radius $ R_0 $ which is centered at the origin, $ 1< p, \; q\le 2$, and $ k_1, k_2 \in C ([0, R_0]; [0, +\infty)).$