Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 379-390, 2019

Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data

Andrea Dall'Aglio, Sergio Segura de León

Received July 6, 2017.   Published online August 6, 2018.

Abstract:  We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $-{\rm div}(a(x,\nabla u))=h(x,u)+\mu$, in $\Omega\subset\R^N$, where the left-hand side is a Leray-Lions operator from $W_0^{1,p} (\Omega)$ into $W^{-1,p'}(\Omega)$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like $|s|^{p-1}$ and $\mu$ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu$.
Keywords:  bounded solution; $p$-Laplacian; renormalized solution; measure data
Classification MSC:  35B45, 35B65, 35J15, 35J25, 35J60, 35J92
DOI:  10.21136/CMJ.2018.0322-17

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Affiliations:   Andrea Dall'Aglio, Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", Piazzale A. Moro 2, I-00185 Roma, Italy, e-mail:; Sergio Segura de León, Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain, e-mail:

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