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We consider a nonlocal eigenvalue problem which arises in the study of stability of point-condensation solutions in the Gierer-Meinhardt system and generalized Gray-Scott system. We give some su cient conditions for stability and instability. The conditions are new and can be applied to the study of stability of single point-condensation solutions.doi:10.1142/s0218127400000979 fatcat:ruoij3omrbgvdmzikcc3nnrpvm