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A Liouville-type theorem for Lane-Emden systems. (English) Zbl 1033.35032
The authors provide a partial positive answer to a well-known conjecture about the nonexistence of positive solutions to Lane-Emden systems below the critical Sobolev hyperbola. The proof is based on a monotonicity argument for suitable transformed functions. It relies on a special form of the Alexandrov-Serrin moving planes method, as well as some refined forms of the maximum principle for elliptic systems that they develop here.
Their result extends the nonexistence result of D. G. de Figueiredo and P. L. Felmer [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. (4) 21, 387–597 (1994; Zbl 0820.35042)].

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B50 Maximum principles in context of PDEs