Differential and Integral Equations

Nonlinear eigenvalue problems for degenerate elliptic systems

Mabel Cuesta and Peter Takáč

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation