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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