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Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential. (English) Zbl 1022.31005
Summary: We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of \(p\)-Laplacian type. If \(p<\gamma <N\) and the right-hand side is a Radon measure with singularity of order \(\gamma \) at \(x_0\in \Omega \), then any supersolution in \(W_{\text{loc}}^{1,p}(\Omega)\) has singularity of order at least \((\gamma -p)/(p-1)\) at \(x_0\). In the proof we exploit a pointwise estimate of \(\mathcal A\)-superharmonic solutions, due to T. Kilpeläinen and J. Malý, which involves Wolff’s potential of Radon’s measure.
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35J60 Nonlinear elliptic equations
35A20 Analyticity in context of PDEs
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References:
[1] G. Díaz and R. Leletier: Explosive solutions of quasilinear elliptic equations: existence and uniqueness. Nonlinear Anal. 20 (1993), 97-125. · Zbl 0793.35028
[2] M. Giaquinta: Multiple Integrals in the Calculus of Variations and Elliptic Systems. Princeton University Press, Princeton, New Jersey, 1983. · Zbl 0516.49003
[3] M. Grillot: Prescribed singular submanifolds of some quasilinear elliptic equations. Nonlinear Anal. 34 (1998), 839-856. · Zbl 0947.35059
[4] J. Heinonen, T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford, 1993. · Zbl 0780.31001
[5] T. Kilpeläinen: Singular solutions to \(p\)-Laplacian type equations. Ark. Mat. 37 (1999), 275-289. · Zbl 1018.35028
[6] T. Kilpeläinen and J. Malý: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa 19 (1992), 591-613. · Zbl 0797.35052
[7] 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. · Zbl 0933.35061
[8] L. Korkut, M. Pašić and D. Žubrinić: Some qualitative properties of solutions of quasilinear elliptic equations and applications. J. Differential Equations 170 (2001), 247-280. · Zbl 0979.35051
[9] J. Leray and J.-L. Lions: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965), 97-107. · Zbl 0132.10502
[10] L. Mou: Removability of singular sets of harmonic maps. Arch. Rational Mech. Anal. 127 (1994), 199-217. · Zbl 1026.58017
[11] L. Simon: Singularities of Geometric Variational Problems. IAS/Park City Math. Ser. Vol. 2. 1992, pp. 183-219.
[12] D. Žubrinić: Generating singularities of solutions of quasilinear elliptic equations. J. Math. Anal. Appl. 244 (2000), 10-16. · Zbl 0947.35064
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