Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential

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We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of pLaplacian type. If p < γ < N and the right-hand side is a Radon measure with singularity of order γ at x0 ∈ Ω, then any supersolution in Wloc1,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.

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