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.


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35J60 Nonlinear elliptic equations
35A20 Analyticity in context of PDEs
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