Publication details
Elliptic equations with nonlinear absorption depending on the solution and its gradient
| Authors | |
|---|---|
| Year of publication | 2015 |
| Type | Article in Periodical |
| Magazine / Source | Proceedings of the London Mathematical Society |
| MU Faculty or unit | |
| Citation | |
| web | https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms/pdv020 |
| Doi | http://dx.doi.org/10.1112/plms/pdv020 |
| Keywords | quasilinear equations;boundary singularities;Radon measures;Borel measures;weak singularities;strong singularities;boundary trace;removability |
| Description | We study positive solutions of equation (E1) -Delta u + u(p)vertical bar del u vertical bar(q) = 0 (0 <= p, 0 <= q <= 2, p + q > 1) and (E-2) -Delta u + u(p) + vertical bar Delta u vertical bar(q) = 0 (p > 1, 1 < q <= 2) in a smooth bounded domain Omega subset of R-N. We obtain a sharp condition on p and q under which, for every positive, finite Borel measure mu on partial derivative Omega, there exists a solution such that u = mu on partial derivative Omega. Furthermore, if the condition mentioned above fails, then any isolated point singularity on partial derivative Omega is removable, namely, there is no positive solution that vanishes on partial derivative Omega everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of partial derivative Omega. In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when p = 0 but not the general case. |