×

zbMATH — the first resource for mathematics

Nonexistence results for classes of elliptic systems. (English) Zbl 1210.35125
Summary: We consider the system \[ -\Delta u=\lambda f(u,v),\;-\Delta v=\lambda g(u,v),\;x\in \Omega ,\;u=0=v,\;x\in \partial \Omega , \] where \(\Omega \) is a ball in \(\mathbb R^N\), \(N\geq 1\) 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.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite