zbMATH — the first resource for mathematics

Non-existence results for semilinear cooperative elliptic systems via moving spheres. (English) Zbl 0962.35054
Nonexistence results for positive (nonnegative) solutions for cooperative elliptic systems of the (vector) form \(-\Delta u= f(u)\) in \(D\), \(u= 0\) on \(\partial D\) are proved via the method of moving spheres. This constitutes a variant of the celebrated moving plane method due to Alexandrov and applied by Serrin, Gidas-Nirenberg and many others to get symmetry results for positive solutions. This approach of moving spheres was used by McCuan and Padilla for obtaining symmetry results, too. The moving sphere method, in the authors’ claim, unifies and simplifies previous work, even for systems. The method of moving planes raises compactness problems when dealing with unbounded domains, a difficulty which can be overcome in this way.
Theorem 1 states that if \(D\subset \mathbb{R}^n\) \((n\geq 3)\) is a bounded starshaped domain and \(f\) is locally Lipschitz and supercritical, there is no positive solution. Some singularities in the \(x\) variable can be allowed. A similar result (Theorem 2) is proved if \(D\) is starshaped with respect to infinity and \(f\) is subcritical (again in a suitable sense). This gives as corollaries nonexistence results by Gidas and Spruck on \(\mathbb{R}^n\) and the half-space, and also for “curved” half-spaces. An interesting monotonicity result (Theorem 3) is very instrumental here. Theorem 4 is an interesting corollary for power nonlinearities and Theorem 5 says that nonnegative solutions to \(-\Delta u= f(u)\) on \(\mathbb{R}^n_+\) with \(u= 0\) on \(\partial\mathbb{R}^n_+\) for some subcritical \(f\)’s depend only on \(x_1\) and are increasing. Some applications to singular problems are also included. In particular, proofs use many subtle comparison arguments and variants of maximum principles.

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B50 Maximum principles in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
[1] Alexandrov, A.D., A characteristic property of the spheres, Ann. mat. pura appl., 58, 303-315, (1962) · Zbl 0107.15603
[2] Bandle, C.; Levine, H.A., On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. amer. math. soc., 316, 595-622, (1989) · Zbl 0693.35081
[3] Berestycki, H., Qualitative properties of semilinear elliptic equations in unbounded domains, Progress in elliptic and parabolic partial differential equations (capri, 1994), Pitman res. notes math. ser., 350, (1996), Longman Harlow, p. 19-42 · Zbl 0887.35054
[4] Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. soc. bras. mat., 22, 1-37, (1991) · Zbl 0784.35025
[5] Berestycki, H.; Caffarelli, L.; Nirenberg, L., Symmetry for elliptic equations in a half space, Boundary value problems for partial differential equations and applications, RMA res. notes appl. math., 29, (1993), Masson Paris, p. 27-42 · Zbl 0793.35034
[6] Berestycki, H.; Caffarelli, L.; Nirenberg, L., Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. pure appl. math., 50, 1089-1111, (1997) · Zbl 0906.35035
[7] H. Berestycki, L. Caffarelli, and, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, preprint, 1997. · Zbl 1079.35513
[8] J. Busca, Symmetry and nonexistence results for Emden-Fowler equations in cones, preprint, 1998. · Zbl 1056.35062
[9] Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure appl. math., XLII, 272-297, (1989) · Zbl 0702.35085
[10] Chen, Wenxiong; Li, Congming, Classification of solutions of some nonlinear elliptic equations, Duke math. J., 63, 615-622, (1991) · Zbl 0768.35025
[11] Dancer, E.N., Some notes on the method of moving planes, Bull. austral. math. soc., 46, 425-434, (1992) · Zbl 0777.35005
[12] Esteban, M.J.; Lions, P.L., Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. royal soc. Edinburgh A, 93, 1-14, (1982) · Zbl 0506.35035
[13] Gidas, B.; Ni, Wei-Ming; Nirenberg, L., Symmetry and related problems via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[14] Gidas, B.; Ni, Wei-Ming; Nirenberg, L., Symmetry of positive solutions of nonlinear equations in \(R\)^n, Math. anal. appl. part A, adv. math. suppl. studies A, 7, 369-402, (1981) · Zbl 0469.35052
[15] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003
[16] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041
[17] Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary value problem, Proc. amer. math. soc., 111, 721-730, (1991) · Zbl 0727.35057
[18] Li, Congming, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. partial differential equations, 16, 585-615, (1991) · Zbl 0741.35014
[19] J. McCuan, Symmetry via spherical reflection, J. Geom. Anal, to appear. · Zbl 1006.53007
[20] Mitidieri, E., A Rellich type identity and applications, Comm. partial differential equations, 18, 125-151, (1993) · Zbl 0816.35027
[21] Mitidieri, E., Nonexistence of positive solutions of semilinear elliptic systems in \(R\)^N, Differential integral equations, 9, 465-479, (1996) · Zbl 0848.35034
[22] Padilla, P., Symmetry properties of positive solutions of elliptic equations on symmetric domains, Appl. anal., 64, 153-169, (1997) · Zbl 0942.35084
[23] Pohovzaev, S.I., Eigenfunctions of the equation δu+λf(u)=0, Soviet math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202
[24] Pucci, P.; Serrin, J., A general variational identity, Indiana univ. J., 35, 681-703, (1986) · Zbl 0625.35027
[25] Serrin, J., A symmetry theorem in potential theory, Arch. rational mech. anal., 43, 304-318, (1971) · Zbl 0222.31007
[26] van der Vorst, R.C.A.M., Variational identities and applications to differential systems, Arch. rational mech. anal., 116, 375-398, (1991) · Zbl 0796.35059
[27] Zou, Henghui, Symmetry of positive solutions of δu+up=0 in \(R\)n, J. differential equations, 120, 46-88, (1995) · Zbl 0844.35028
[28] Zou, Henghui, Slow decay and the Harnack inequality for positive solutions of δu+up=0 in \(R\)n, Differential integral equations, 8, 1355-1368, (1995) · Zbl 0849.35028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.