Differential and Integral Equations

Nonlinear eigenvalue problems for degenerate elliptic systems

Mabel Cuesta and Peter Takáč

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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

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

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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