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A Liouville-type theorem for elliptic systems. (English) Zbl 0820.35042
The authors consider the system $$- \triangle u = v^ \alpha$$, $$- \triangle v = u^ \beta$$ in the whole of $$\mathbb{R}^ N$$, $$N \geq 3$$. The question is to determine for which values of the exponents $$\alpha$$ and $$\beta$$ the only nonnegative solution $$(u,v)$$ is the trivial one.
Theorem: If $$\alpha > 0, \beta > 0$$ are such that $$\alpha, \beta \leq (N + 2)/(N - 2)$$, but not both are equal to $$(N + 2)/(N - 2)$$, then the only nonnegative $$C^ 2$$ solution of the system in the whole $$\mathbb{R}^ N$$ is the trivial one. If $$\alpha = \beta = (N + 2)/(N - 2)$$, then $$u$$ and $$v$$ are radially symmetric with respect to some point of $$\mathbb{R}^ N$$.
Reviewer: O.John (Praha)

##### MSC:
 35J45 Systems of elliptic equations, general (MSC2000) 35J60 Nonlinear elliptic equations
##### Keywords:
superlinear elliptic systems; Liouville-type theorem
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##### References:
  H. Berestycki - L. Nirenberg , On the method of moving planes and the sliding method . Bol. Soc. Brasil. Mat. 22 ( 1991 ), 1 - 22 . MR 1159383 | Zbl 0784.35025 · Zbl 0784.35025 · doi:10.1007/BF01244896  Ph. Clément - D.G. De Figueiredo - E. Mitidieri , Positive solutions of semilinear elliptic systems . Comm. Partial Differential Equations 17 ( 1992 ), 923 - 940 . MR 1177298 | Zbl 0818.35027 · Zbl 0818.35027 · doi:10.1080/03605309208820869  L. Caffarelli - B. Gidas - J. Spruck , Asymptotic Symmetry and local behavior of Semilinear Elliptic Equations with Critical Sobolev Growth . Comm. Pure App. Math. , XLII ( 1989 ), 271 - 297 . MR 982351 | Zbl 0702.35085 · Zbl 0702.35085 · doi:10.1002/cpa.3160420304  W. Chen - C. Li , Classification of solutions of some nonlinear elliptic equations . Duke Math. J. 63 ( 1991 ), 615 - 622 . Article | MR 1121147 | Zbl 0768.35025 · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8 · minidml.mathdoc.fr  D.G. De Figueiredo - P.L. Felmer , On Superquadratic Elliptic Systems . To appear in Trans. Amer. Math. Soc. MR 1214781 | Zbl 0799.35063 · Zbl 0799.35063 · doi:10.2307/2154523  D.G. De Figueiredo - E. Mitidieri , Maximum Principles for Linear Elliptic Systems . Rend. Ist. Mat. Univ. Trieste XXII ( 1990 ), 36 - 66 . MR 1210477 | Zbl 0793.35011 · Zbl 0793.35011  J. Hulshof - R. Van Der Vorst , Differential Systems with Strongly Indefinite Variational Structure . J. Fatl. Anal , vol. 114 n^\circ 1 ( 1993 ), 32 - 58 . MR 1220982 | Zbl 0793.35038 · Zbl 0793.35038 · doi:10.1006/jfan.1993.1062  B. Gidas , Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations . In: Nonlinear Partial Differential Equations in Engineering and Applied Sciences . Editors R. Sternberg, A. Kalinovski and J. Papadakis. Marcel Dekker Inc. , 1980 . MR 577096 | Zbl 0444.35038 · Zbl 0444.35038  B. Gidas - W.M. Ni - L. Nirenberg , Symmetry and related properties via the maximum principle . Comm. Math. Phys. 68 ( 1979 ), 209 - 243 . Article | MR 544879 | Zbl 0425.35020 · Zbl 0425.35020 · doi:10.1007/BF01221125 · minidml.mathdoc.fr  B. Gidas - J. Spruck , A priori bounds for positive solutions of nonlinear elliptic equations , Comm. Partial Differential Equations 6 ( 1981 ), 883 - 901 . MR 619749 | Zbl 0462.35041 · Zbl 0462.35041 · doi:10.1080/03605308108820196  B. Gidas - J. Spruck , Global and local behavior of positive solutions of nonlinear elliptic equations . Comm. Pure App. Math. 34 ( 1981 ), 525 - 598 . MR 615628 | Zbl 0465.35003 · Zbl 0465.35003 · doi:10.1002/cpa.3160340406  Qing Jie , A priori estimates for positive solutions of semilinear elliptic systems , J. Partial Differential Equations 1 ( 1988 ), 61 - 70 . MR 985447 | Zbl 0682.35041 · Zbl 0682.35041  E. Mitidieri , A Rellich type identity and applications . To appear in Comm. Partial Differential Equations . MR 1211727 | Zbl 0816.35027 · Zbl 0816.35027 · doi:10.1080/03605309308820923  M.H. Protter - H.F. Weinberger , Maximum principles in differential equations , Prentice Hall ( 1967 ). MR 219861 | Zbl 0153.13602 · Zbl 0153.13602  M.A. Souto , Sobre a existência de soluções positivas para sistemas cooperativos não lineares . PhD thesis, Unicamp ( 1992 ).
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