## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 11/12 (2010), 1117-1138.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 11/12 (2010), 1117-1138.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019076

**Mathematical Reviews number (MathSciNet)**

MR2742481

**Zentralblatt MATH identifier**

1240.35193

**Subjects**

Primary: 35J70: Degenerate elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H12

#### Citation

Cuesta, Mabel; Takáč, Peter. Nonlinear eigenvalue problems for degenerate elliptic systems. Differential Integral Equations 23 (2010), no. 11/12, 1117--1138. https://projecteuclid.org/euclid.die/1356019076