In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equationα 2 stands for the fractional Laplacian operator, f ∈ C(Ω × R, R) may be sign changing and λ is a positive parameter. We will prove that there exists λ * > 0 such that the problem has at least two positive solutions for each λ ∈ (0 , λ * ). In addition, the concentration behavior of the solutions are investigated.2010 Mathematics Subject Classification. 35J60, 47J30.

In Pseudomonas aeruginosa, the second messenger cyclic-di-GMP and Gac/Rsm signaling pathways are associated with the transition from acute to chronic infection. Therefore, identifying the molecular mechanisms that govern lifestyle choice in bacteria is very important. Here, we identified a novel cyclic-di-GMP modulator, PrtR, which was shown to repress pyocin production by inhibition of PrtN and activate the Type III secretion system (T3SS) through PtrB. Compared to a wild-type strain or a prtN mutant, the prtR/prtN double mutant exhibited a wrinkly colony and hyperbiofilm phenotype as well as an increase in intracellular c-di-GMP levels. Interestingly, a diguanylate cyclase (DGC) gene, siaD, was repressed by PrtR. Further experiments revealed that PrtR directly interacts with SiaD, and facilitates the accumulation of c-di-GMP in cells. We also demonstrated that PrtR regulates the activity of Gac/Rsm system, thus affecting expression of the T3SS, type VI secretion system (T6SS) and the formation of biofilm. Taken together, the present findings indicate that PrtR, as a c-di-GMP modulator, plays key roles for the adaptation to the opportunistic infection of P. aeruginosa. Additionally, this study revealed a novel mechanism for PrtR-mediated regulation of the lifestyle transition via the Gac/Rsm and c-di-GMP signaling networks.

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