The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations: the power like case

Abstract

We study the explosive expansion near the boundary of the large solutions of the equation

$$-\Delta_{p}u+u^{m}=f \quad{\rm in} \Omega$$

where \({\Omega}\) is an open bounded set of \({\mathbb{R}^{N}}\) , N > 1, with adequately smooth boundary, m > p−1 > 0, and f is a continuous nonnegative function in \({\Omega}\) . Roughly speaking, we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite, but it goes to infinity as m goes to p−1. For illustrative choices of the sources, we prove that the expansion consists of two possible geometrical and nongeometrical parts. For low explosive sources, the nongeometrical part does not exist, and all coefficients depend on the diffusion and the geometry of the domain. For high explosive sources, there are coefficients, relative to the nongeometrical part, independent on \({\Omega}\) and the diffusion. In this case, the geometrical part cannot exist, and we say then that the source is very high explosive. We emphasize that low or high explosive sources can cause different geometrical properties in the expansion for a given interior structure of the differential operator. This paper is strongly motivated by the applications, in particular by the non-Newtonian fluid theory where p ≠ 2 involves rheological properties of the medium.

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Correspondence to G. Díaz.

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S.A. was partially supported by Fondecyt Grant No. 11110482, USM Grant No. 121210 and Programa Basal, CMM, U. de Chile. G.D. is supported by the projects MTM 2008-06208 of DGISGPI (Spain) and the Research Group MOMAT (Ref. 910480) from Banco Santander and UCM. The work of J.M.R. has been done in the framework of project MTM2011-22658 of the Spanish Ministry of Science and Innovation and the Research Group MOMAT (Ref. 910480) supported by Banco Santander and UCM.

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Alarcón, S., Díaz, G. & Rey, J.M. The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations: the power like case. Z. Angew. Math. Phys. 64, 659–677 (2013). https://doi.org/10.1007/s00033-012-0253-5

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Mathematics Subject Classification

  • 35B40
  • 35J25
  • 35J65

Keywords

  • Large solutions
  • Asymptotic behavior
  • Sub- and super-solutions