Abstract
We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of p-Laplacian type. If p < γ < N and the right-hand side is a Radon measure with singularity of order γ at x0A ∈ ω, then any supersolution in W1,p(ω) has singularity of order at least (γ−p)/(p−1) at x0. In the proof we exploit a pointwise estimate of A-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff's potential of Radon's measure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Díaz and R. Leletier: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Anal. 20 (1993), 97-125.
M. Giaquinta: Multiple Integrals in the Calculus of Variations and Elliptic Systems. Princeton University Press, Princeton, New Jersey, 1983.
M. Grillot: Prescribed singular submanifolds of some quasilinear elliptic equations. Nonlinear Anal. 34 (1998), 839-856.
J. Heinonen, T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford, 1993.
T. Kilpeläinen: Singular solutions to p-Laplacian type equations. Ark. Mat. 37 (1999), 275-289.
T. Kilpeläinen and J. Malý: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa 19 (1992), 591-613.
L. Korkut, M. Pašić and D. Žubrinić: Control of essential infimum and supremum of solutions of quasilinear elliptic equations. C. R. Acad. Sci. Paris t. 329, Série I (1999), 269-274.
L. Korkut, M. Paši? and D. Žubrinić: Some qualitative properties of solutions of quasilinear elliptic equations and applications. J. Differential Equations 170 (2001), 247-280.
J. Leray and J.-L. Lions: Quelques résultats de Visik sur les probl_emes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965), 97-107.
L. Mou: Removability of singular sets of harmonic maps. Arch. Rational Mech. Anal. 127 (1994), 199-217.
L. Simon: Singularities of Geometric Variational Problems. IAS/Park City Math. Ser. Vol. 2. 1992, pp. 183-219.
D. Žubrinć: Generating singularities of solutions of quasilinear elliptic equations. J. Math. Anal. Appl. 244 (2000), 10-16.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Žubrinić, D. Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential. Czechoslovak Mathematical Journal 53, 429–435 (2003). https://doi.org/10.1023/A:1026247706484
Issue Date:
DOI: https://doi.org/10.1023/A:1026247706484