Boundary singularities of solutions of N-harmonic equations with absorption

Abstract

We study the boundary behaviour of solutions u of − N u + |u| q−1 u = 0 in a bounded smooth domain Ω ⊂ R N subject to the boundary condition u = 0 except at one point, in the range q > N − 1. We prove that if q 2N − 1 such an u is identically zero, while, if N − 1 < q < 2N − 1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.

Boundary Harnack Inequality and a Priori Estimates of Singular Solutions of Quasilinear Elliptic Equations

“…The next result, which extends a previous result in [2], made more precise the statement of Theorem 2.13. Proof.…”

“…If p > 2 and u admits no critical point in Ω, L is uniformly elliptic ([5], [2] for details in a similar situation). Thus the strong maximum principles holds and w is locally zero.…”

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