Nonlinear eigenvalue problems for degenerate elliptic systems

Abstract

The following nonlinear eigenvalue problem for a pair of real parameters $(\lambda,\mu)$ is studied: $$ \begin{cases} - \Delta_p u = \lambda\, a(x)\, |u|^{\alpha_1} |v|^{\beta_1 - 1} v & \mbox{ in }\, \Omega; \\ - \Delta_q v = \mu\, b(x)\, |v|^{\alpha_2} |u|^{\beta_2 - 1} u & \mbox{ in }\, \Omega; \\ u = v = 0 & \mbox{ on }\, \partial\Omega. \end{cases} $$ Here, $p,q\in (1,\infty)$ are given numbers, $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with a $C^2$-boundary, $a,b\in L^{\infty}(\Omega)$ are given functions, both assumed to be strictly positive on compact subsets of $\Omega$, and the coefficients $\alpha_i, \beta_i$ are nonnegative numbers satisfying either the conditions $ \alpha_1 + \beta_1 = p-1 \,\mbox{ and }\, \alpha_2 + \beta_2 = q-1, $ or the condition $$ (p-1 - \alpha_1) (q-1 - \alpha_2) = \beta_1\beta_2. $$ A {\em smooth curve} of pairs $(\lambda,\mu)$ in $(0,\infty)\times (0,\infty)$ is found for which the quasilinear elliptic system possesses a solution pair $(u,v)$ consisting of nontrivial, nonnegative functions $u\in W_0^{1,p}(\Omega)$ and $v\in W_0^{1,q}(\Omega)$. Key roles in the proof are played by the strong comparison principle and a nonlinear Kreĭn-Rutman theorem obtained by the authors in earlier works. The main result is applied to some quasilinear elliptic systems related to the above system.