Abstract
We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as \(g(u)|\nabla u|^q\), where \(1<q<2\) and g(s) is a continuous function. Data belong to \(L^m(\Omega )\) with \(1\le m <\frac{N}{2}\) as well as measure data instead of \(L^1\)-data, so that unbounded solutions are expected. Our aim is, given \(1\le m<\frac{N}{2}\) and \(1<q<2\), to find the suitable behaviour of g close to infinity which leads to existence for our problem. We show that the presence of g has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either g(s) is constant or \(q=2\).
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The third author has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER, under Project PGC2018–094775–B–I00.
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Latorre, M., Magliocca, M. & Segura de León, S. Regularizing effects concerning elliptic equations with a superlinear gradient term. Rev Mat Complut 34, 297–356 (2021). https://doi.org/10.1007/s13163-020-00353-z
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DOI: https://doi.org/10.1007/s13163-020-00353-z
Keywords
- Quasilinear elliptic equations
- Gradient term with superlinear growth
- Renormalized solutions
- Measure data