Abstract
Given a smooth domain \({\Omega\subset\mathbb{R}^N}\) such that \({0 \in \partial\Omega}\) and given a nonnegative smooth function ζ on ∂Ω, we study the behavior near 0 of positive solutions of −Δu = u q in Ω such that u = ζ on ∂Ω\{0}. We prove that if \({\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}\) , then \({u(x)\leq C |x|^{-\frac{2}{q-1}}}\) and we compute the limit of \({|x|^{\frac{2}{q-1}} u(x)}\) as x → 0. We also investigate the case \({q= \frac{N+1}{N-1}}\) . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Aviles P.: Local behavior of solutions of some elliptic equations. Commun. Math. Phys. 108, 177–192 (1987)
Bandle C., Benguria R.: The Brezis–Nirenberg problem on S 3. J. Differ. Equ. 178, 264–279 (2002)
Bandle C., Wei J.: Multiple clustered layer solutions for semilinear elliptic problems on S n. Commun. Partial Differ. Equ. 33, 613–635 (2008)
Bellman R.: Stability Theory of Differential Equations. McGraw-Hill Book Company, Inc., New York (1953)
Bidaut-Véron M.-F., Bouhar M.: On characterization of solutions of some nonlinear differential equations and applications. SIAM J. Math. Anal. 25, 859–875 (1994)
Bidaut-Véron M.-F., Raoux T.: Asymptotics of solutions of some nonlinear elliptic systems. Commun. Partial Differ. Equ. 21, 1035–1086 (1996)
Bidaut-Véron M.-F., Vivier L.: An elliptic semilinear equation with source term involving boundary measures: the subcritical case. Rev. Mat. Iberoam. 16, 477–513 (2000)
Bidaut-Véron M.-F., Ponce A.C., Véron L.: Boundary singularities of positive solutions of some nonlinear elliptic equations. C. R. Math. Acad. Sci. Paris 344, 83–88 (2007)
Bidaut-Véron M.-F., Jazar M., Véron L.: Separable solutions of some quasilinear equations with source reaction. J. Differ. Equ. 244, 274–308 (2008)
Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Brezis H., Peletier L.A.: Elliptic equations with critical exponent on spherical caps of S 3. J. Anal. Math. 98, 279–316 (2006)
Chen W., Wei J.: On the Brezis–Nirenberg problem on S 3, and a conjecture of Bandle–Benguria. C. R. Math. Acad. Sci. Paris 341, 153–156 (2005)
Dancer E.N.: Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46, 425–434 (1992)
del Pino M., Musso M., Pacard F.: Boundary singularities for weak solutions of semilinear elliptic problems. J. Funct. Anal. 253, 241–272 (2007)
Dynkin E.B., Kuznetsov S.E.: Superdiffusions and removable singularities for quasilinear partial differential equations. Commun. Pure Appl. Math. 49, 125–176 (1996)
Dynkin E.B., Kuznetsov S.E.: Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary. Trans. Am. Math. Soc. 350, 4521–4552 (1998)
Fabbri J., Véron L.: Singular boundary value problems for nonlinear elliptic equations in nonsmooth domains. Adv. Differ. Equ. 1, 1075–1098 (1996)
Gidas B., Spruck J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)
Gmira A., Véron L.: Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64, 271–324 (1991)
Kwong M.K.: Uniqueness results for Emden–Fowler boundary value problems. Nonlinear Anal. 16, 435–454 (1991)
Kwong M.K., Li Y.: Uniqueness of radial solutions of semilinear elliptic equations. Trans. Am. Math. Soc. 333, 339–363 (1992)
Le Gall J.-F.: The Brownian snake and solutions of Δu = u 2 in a domain. Probab. Theory Relat Fields 102, 393–432 (1995)
Lions P.-L.: Isolated singularities in semilinear problems. J. Differ. Equ. 38, 441–450 (1980)
Marcus M., Véron L.: The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case. Arch. Ration. Mech. Anal. 144, 201–231 (1998a)
Marcus M., Véron L.: The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case. J. Math. Pures Appl. 77, 481–524 (1998b)
Marcus M., Véron L.: Removable singularities and boundary traces. J. Math. Pures Appl. 80, 879–900 (2001)
Padilla P.: Symmetry properties of positive solutions of elliptic equations on symmetric domains. Appl. Anal. 64, 153–169 (1997)
Poláčik P., Quittner P., Souplet P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math. J. 139, 555–579 (2007)
Véron L.: Singularités éliminables d’équations elliptiques non linéaires. J. Differ. Equ. 41, 87–95 (1981)
Author information
Authors and Affiliations
Corresponding authors
Additional information
Commmunicated by H.Brezis.
Rights and permissions
About this article
Cite this article
Bidaut-Véron, MF., Ponce, A.C. & Véron, L. Isolated boundary singularities of semilinear elliptic equations. Calc. Var. 40, 183–221 (2011). https://doi.org/10.1007/s00526-010-0337-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0337-z