Differential and Integral Equations

Nonexistence results for classes of elliptic systems

R. Shivaji and Jinglong Ye

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We consider the system \[ -\Delta u = \lambda f(u,v); \, x \in \Omega \] \[ -\Delta v = \lambda g(u,v); \, x \in \Omega \] \[ u = 0 = v; \, x \in \partial\Omega, \] where $\Omega$ is a ball in $ R^{N}, N \geq 1$ and $\partial\Omega$ is its boundary, $\lambda $ is a positive parameter, and $f$ and $g$ are smooth functions that are negative at the origin (semipositone system) and satisfy certain linear growth conditions at infinity. We establish nonexistence of positive solutions when $\lambda$ is large. Our proofs depend on energy analysis and comparison methods.

Article information

Differential Integral Equations, Volume 20, Number 8 (2007), 927-938.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 35J65: Nonlinear boundary value problems for linear elliptic equations


Shivaji, R.; Ye, Jinglong. Nonexistence results for classes of elliptic systems. Differential Integral Equations 20 (2007), no. 8, 927--938. https://projecteuclid.org/euclid.die/1356039364

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