×

On unique continuation principles for some elliptic systems. (English) Zbl 1511.35147

Summary: In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci’s extremal operators, being new even for scalar equations.

MSC:

35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12, 623-727 (1959) · Zbl 0093.10401
[2] Alinhac, S.; Baouendi, M. S., Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities, Am. J. Math., 102, 1, 179-217 (1980) · Zbl 0425.35098
[3] Armstrong, S. N.; Silvestre, L., Unique continuation for fully nonlinear elliptic equations, Math. Res. Lett., 18, 5, 921-926 (2011) · Zbl 1241.35060
[4] Bonheure, D.; Moreira dos Santos, E.; Ramos, M., Symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems, J. Funct. Anal., 264, 1, 62-96 (2013) · Zbl 1278.35069
[5] Caffarelli, L.; Crandall, M. G.; Kocan, M.; Swiech, A., On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49, 4, 365-397 (1996) · Zbl 0854.35032
[6] Caffarelli, L. A.; Cabré, X., Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43 (1995), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0834.35002
[7] Caffarelli, L. A.; Friedman, A., The free boundary in the Thomas-Fermi atomic model, J. Differ. Equ., 32, 3, 335-356 (1979) · Zbl 0408.35083
[8] Caffarelli, L. A.; Friedman, A., Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations, J. Differ. Equ., 60, 3, 420-433 (1985) · Zbl 0593.35047
[9] Carleman, T., Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astron. Fys., 26, 17, 9 (1939) · Zbl 0022.34201
[10] Colombini, F.; Grammatico, C., Some remarks on strong unique continuation for the Laplace operator and its powers, Commun. Partial Differ. Equ., 24, 5-6, 1079-1094 (1999) · Zbl 0928.35041
[11] Colombini, F.; Koch, H., Strong unique continuation for products of elliptic operators of second order, Trans. Am. Math. Soc., 362, 1, 345-355 (2010) · Zbl 1184.35091
[12] Dalmasso, R., Existence and uniqueness of positive radial solutions for the Lane-Emden system, Nonlinear Anal., 57, 3, 341-348 (2004) · Zbl 1069.34032
[13] Farina, A., Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems, Discrete Contin. Dyn. Syst., 35, 12, 5869-5877 (2015) · Zbl 1335.35054
[14] Garofalo, N.; Lin, F.-H., Monotonicity properties of variational integrals, \( A_p\) weights and unique continuation, Indiana Univ. Math. J., 35, 2, 245-268 (1986) · Zbl 0678.35015
[15] Garofalo, N.; Lin, F.-H., Unique continuation for elliptic operators: a geometric-variational approach, Commun. Pure Appl. Math., 40, 3, 347-366 (1987) · Zbl 0674.35007
[16] Gazzola, F.; Grunau, H.-C.; Sweers, G., Positivity preserving and nonlinear higher order elliptic equations in bounded domains, (Polyharmonic Boundary Value Problems. Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics, vol. 1991 (2010), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1239.35002
[17] Han, Q., Singular sets of solutions to elliptic equations, Indiana Univ. Math. J., 43, 3, 983-1002 (1994) · Zbl 0817.35020
[18] Hörmander, L., The Analysis of Linear Partial Differential Operators. IV: Fourier Integral Operators, Classics in Mathematics (2009), Springer-Verlag: Springer-Verlag Berlin, Reprint of the 1994 edition · Zbl 1178.35003
[19] Jerison, D.; Kenig, C. E., Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math. (2), 121, 3, 463-494 (1985), With an appendix by E.M. Stein · Zbl 0593.35119
[20] Koch, H.; Tataru, D., Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Commun. Pure Appl. Math., 54, 3, 339-360 (2001) · Zbl 1033.35025
[21] Koike, S.; Świech, A., Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339, 2, 461-484 (2007) · Zbl 1387.35243
[22] Li, Y. Y., Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differ. Equ., 83, 2, 348-367 (1990) · Zbl 0748.35013
[23] Lin, F.-H., Nodal sets of solutions of elliptic and parabolic equations, Commun. Pure Appl. Math., 44, 3, 287-308 (1991) · Zbl 0734.58045
[24] Mitidieri, E., A Rellich type identity and applications, Commun. Partial Differ. Equ., 18, 1-2, 125-151 (1993) · Zbl 0816.35027
[25] Montaru, A.; Sirakov, B.; Souplet, P., Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems, Arch. Ration. Mech. Anal., 213, 1, 129-169 (2014) · Zbl 1298.35060
[26] Montenegro, M., The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 29, 1, 193-229 (2000) · Zbl 0956.35097
[27] Moreira dos Santos, E.; Nornberg, G., Symmetry properties of positive solutions for fully nonlinear elliptic systems, J. Differ. Equ., 269, 4175-4191 (2020) · Zbl 1440.35095
[28] Protter, M. H., Unique continuation for elliptic equations, Trans. Am. Math. Soc., 95, 81-91 (1960) · Zbl 0094.07901
[29] Quittner, P.; Souplet, P., Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44, 4, 2545-2559 (2012) · Zbl 1255.35111
[30] Rüland, A., Unique continuation for sublinear elliptic equations based on Carleman estimates, J. Differ. Equ., 265, 11, 6009-6035 (2018) · Zbl 1397.35047
[31] Saldaña, A.; Tavares, H., Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions, J. Differ. Equ., 265, 12, 6127-6165 (2018) · Zbl 1426.35115
[32] Sirakov, B., Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE, Int. Math. Res. Not., 24, 7457-7482 (2018) · Zbl 1477.35087
[33] Soave, N.; Terracini, S., The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem, Adv. Math., 334, 243-299 (2018) · Zbl 1395.35009
[34] Soave, N.; Terracini, S., The nodal set of solutions to some elliptic problems: singular nonlinearities, J. Math. Pures Appl., 9, 128, 264-296 (2019) · Zbl 1422.35083
[35] Soave, N.; Weth, T., The unique continuation property of sublinear equations, SIAM J. Math. Anal., 50, 4, 3919-3938 (2018) · Zbl 1395.35057
[36] Troy, W. C., Symmetry properties in systems of semilinear elliptic equations, J. Differ. Equ., 42, 3, 400-413 (1981) · Zbl 0486.35032
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.