Existence of positive solutions to -Δu = Q |u|^{p-2}u on Z^d, half-space, and quadrant is established in Sobolev super-critical (α,p) regions and ruled out in Serrin sub-critical regions, with an open intermediate region in full and half spaces.
Rabinowitz: Minimax methods in critical point theory with applications to differential equations,CBMS Reg
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On positive solutions of Lane-Emden equations on the integer lattice graphs
Existence of positive solutions to -Δu = Q |u|^{p-2}u on Z^d, half-space, and quadrant is established in Sobolev super-critical (α,p) regions and ruled out in Serrin sub-critical regions, with an open intermediate region in full and half spaces.