Regularizing effects concerning elliptic equations with a superlinear gradient term. (English) Zbl 1471.35145
Summary: 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 \(q=2\).
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
quasilinear elliptic equation; gradient term with superlinear growth; renormalized solutions; measure dataReferences:
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