Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist? Thanks! 

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The only such functions are $0$. Compact support implies $$\int_{\Omega} \Delta u = 0.$$ This along with the subsolution hypothesis means that $\Delta u = 0$. Any compactly supported harmonic function is identically zero by analyticity. 

